Download - GENERAL SURVEYING
GHEORGHE M. T. RĂDULESCU
GENERAL TOPOGRAPHY
LECTURE NOTES
2002
FOREWORD
The presented lecture notes are addressed to the students of the Faculty of Mineral
Resources and Environment, to those who attend a form of undergraduate or postgraduate
specialization, to all those who in their design or execution activity find themselves in
situations that require the help of topographic methods and instruments in order to solve
the technical problems related with implementing an investment.
This course represents the general part of TOPOGRAPHY, the direct side by
which planimetric and leveling terrain surfaces are measured in order to be represented
on topographic plans.
Based on an experience of more than 20 years, I have elaborated this work
starting from what I have learnt in the Faculty of Geodesy from Bucharest, from the
prominent specialty teachers: the regretted N. Cristescu, M. Sebastian-Taub, M. Neamtu,
E. Ulea, from my lifetime mentor Mr. Prof. Dr. Eng. Vasile Ursea, then passing to the
collaboration within the specialty department in the Polytechnic Institute from Cluj-
Napoca (during the years 1980-1985) together with the colleagues Ms. Viorica Balan and
Mr. Gheorghe Bendea, and then continuing within the institution where I am working in
the present.
Being addressed also to those that chose a form pf specialization with reduced
frequency (Distance Learning or Without Frequency) I have tried to give this work a
didactical, explicit character, each relation being deduced starting from solutions in
which the interrelation of the processed elements is presented.
The Author
II
TABLE OF CONTENTS
FOREWORD....................................................................................................................II
TABLE OF CONTENTS................................................................................................III
CHAPTER I.......................................................................................................................1
1.1. TOPOGRAPHY – THE SCIENCE OF TERRESTRIAL MEASUREMENTS.......11.1.a. THE ACTIVITY SPHERE OF TERRESTRIAL MEASUREMENTS...................11.1.b. THE OBJECT AND THE TOPOGRAPHIC APPLICATIONS IN CONSTRUCTIONS AND MINING..............................................................................3
1.2. THE PRINCIPLES OF THE TECHNICAL SCIENCE IN TOPOGRAPHY..........61.3. LENGTH AND SURFACE MEASURING UNITS................................................9
1.3.1. LENGTH-MEASURING UNITS........................................................................91.3.2. SURFACE-MEASURING UNITS....................................................................11
CHAPTER II – THE SHAPE AND DIMENSIONS OF EARTH, PROJECTIONS, REPRESENTATIONS....................................................................................................17
2.1. THE GENERAL SHAPE OF THE EARTH..........................................................172.2. THE DIMENSIONS OF THE EARTH..................................................................182.3. CARTOGRAPHIC PROJECTIONS. OVERVIEW. CLASSIFICATIONS..........192.4. CARTOGRAPHIC PROJECTIONS – GENERAL PRINCIPLES........................212.5. PROJECTION SYSTEMS USED ALONG TIME IN OUR COUNTRY.............23
CHAPTER III – TOPOGRAPHIC ELEMENTS OF THE TERRAIN.....................26
3.1. TOPOGRAPHIC SURFACE, DETAILS, GEOMETRIZING THE TERRAIN, CHARACTERISTIC POINTS......................................................................................26
PROJECTIONS, MAPS, PLANS................................................................................27TOPOGRAPHIC ELEMENTS OF THE TERRAIN...................................................29PLANIMETRIC AND LEVELING TOPOGRAPHIC SURVEYS, INTRODUCTORY ELEMENTS................................................................................................................36
CHAPTER IV – ERROR ANALYSIS IN TERRESTRIAL MEASUREMENTS....38
4.1. MEASUREMENT CLASSIFICATION................................................................384.2. NOTIONS CONCERNING ERRORS...................................................................404.3. PRESENTING MEASUREMENT RESULTS......................................................43
CHAPTER V – TOPOGRAPHIC INSTRUMENTS...................................................45
5.1. STUDYING THEODOLITES................................................................................45MAIN AXES AND PARTS OF A THEODOLITE......................................................48THE COMPONENTS OF A THEODOLITE.............................................................50USING THE THEODOLITE......................................................................................56VERIFYING AND RECTIFYING THEODOLITES...................................................60MEASURING ANGLES WITH THE THEODOLITE................................................63INSTRUMENTS FOR DIRECT MEASUREMENT OF DISTANCES........................66OPERATIONS ON THE DIRECT MEASUREMENT OF DISTANCES....................67
III
CORRECTIONS APPLIED TO LENGTHS MEASURED WITH STEEL TAPES.....68THE PRECISION OF DIRECT MEASUREMENT OF DISTANCES........................71ELECTRONIC DEVICES FOR MEASURING DISTANCES....................................71GEOMETRIC LEVELING DEVICES........................................................................72LEVELING DEVICES WITH TELESCOPE..............................................................74VERIFYING AND RECTIFYING LEVELING DEVICES..........................................75TACHEOMETRIC DEVICES....................................................................................77SELFREDUCING TACHEOMETERS WITH REFRACTION OR DIVORCED IMAGE.......................................................................................................................83OPTICAL TELEMETERS..........................................................................................87PARALLACTIC MEASUREMENT OF DISTANCES................................................88TRIGONOMETRIC METHODS FOR MEASURING DISTANCES..........................90INSTRUMENTS AND DEVICES FOR TRANSMITTING POINTS ON THE VERTICAL.................................................................................................................91
CHAPTER VI – PLANIMETRIC SURVEYS..............................................................92
PLANIMETRIC CONTROL NETWORKS.................................................................94THE GEODETIC CONTROL NETWORK – THE GEODETIC CONTROL BASIS..94THE STATE GEODETIC TRANGULATION NETWORK.........................................95LOCAL CONTROL NETWORKS..............................................................................98DESIGNATING AND SIGNALING THE POINTS OF THE PLANIMETRIC CONTROL NETWORK............................................................................................100THE TOPOGRAPHIC DESCRIPTION OF POINTS (THE MARKING FILE OF THE TOPOGRAPHIC POINT)...............................................................................104COMPUTING THE COORDINATES OF CONTROL NETWORKS.......................105CLASSIFICATION OF TRAVERSES......................................................................111
DESIGNING PLANIMETRIC TRAVERSES............................................................112FIELD OPERATIONS.............................................................................................113COMPUTATIONAL OPERATIONS........................................................................115
SURVEY OF PLANIMETRIC DETAILS..................................................................125
CHAPTER VII – LEVELING SURVEYS..................................................................128
THE LEVELING.........................................................................................................128HEIGHTS, LEVEL SURFACES................................................................................128THE EFFECT OF THE INFLUENCE OF THE EARTH CURVATURE AND THE ATMOSPHERIC REFRACTION...............................................................................129LEVELING TYPES....................................................................................................131LEVELING NETWORKS..........................................................................................132DESIGNATING AND SIGNALING LEVELING POINTS......................................134GEOMETRIC LEVELING.........................................................................................135MIDDLE GEOMETRIC LEVELING.........................................................................135END GEOMETRIC LEVELING................................................................................137MIDDLE GEOMETRIC LEVELING TRAVERSES.................................................138CLASSIFICATION OF GEOMETRIC LEVELING TRAVERSES..........................139MIDDLE GEOMETRIC LEVELING TRAVERSE SUPPORTED AT BOTH ENDS.....................................................................................................................................140COMPUTING THE TRAVERSE...............................................................................141
IV
COMPUTING THE LEVELING TRAVERSE IN CIRCUIT....................................143COMPUTING THE FLOATING LEVELING TRAVERSE.....................................143COMPUTING LEVELING NETWORKS..................................................................143LEVELING SURVEY OF SURFACES THROUGH GEOMETRIC LEVELING....143LEVELING RADIATION..........................................................................................143THE METHOD OF SQUARES..................................................................................145SURFACE LEVELING THROUGH SMALL SQUARES........................................145SURFACE LEVELING THROUGH LARGE SQUARES........................................147THE PRECISION OF GEOMETRIC LEVELING.....................................................149SURFACE LEVELING THROUGH PROFILES.......................................................152TRIGONOMETRIC LEVELING...............................................................................154TRIGONOMETRIC LEVELING TRAVERSES........................................................157TRIGONOMETRIC LEVELING RADIATION........................................................159TACHEOMETRIC LEVELING.................................................................................160GRAPHICAL PRECISION OF TOPOGRAPHIC PLANS........................................161CLASSIFICATION OF MAPS AND PLANS............................................................162TOPOGRAPHIC SYMBOLS.....................................................................................162LEVELING SYMBOLS..............................................................................................163
CHAPTER VIII – PLANS AND MAPS......................................................................165
8.1. THE ELEMENTS OF PLANS AND MAPS.......................................................165DEFINITIONS.........................................................................................................165SCALES....................................................................................................................165THE GRAPHICAL PRECISION OF TOPOGRAPHIC PLANS..............................167CLASSIFICATION OF MAPS AND PLANS...........................................................167TOPOGRAPHIC SYMBOLS....................................................................................167LEVELING SYMBOLS.............................................................................................168
8.2. USING MAPS AND PLANS...............................................................................1711. DETERMINING THE GEOGRAPHIC COORDINATES OF A POINT ON THE MAP.........................................................................................................................1712. DETERMINING THE CARTESIAN COORDINATES OF A POINT ON THE MAP/PLAN..............................................................................................................1723. REPEATING A POINT ON THE MAP/PLAN THROUGH CARTESIAN COORDINATES......................................................................................................1734. DETERMINING THE HORIZONTAL DISTANCE BETWEEN TWO POINTS ON THE MAP/PLAN......................................................................................................1745. DETERMINING THE ORIENTATION OF A DIRECTION ON THE MAP/PLAN.................................................................................................................................1756. THE ORIENTATION IN THE FIELD OF MAPS AND PLANS..........................1757. DETERMINING SURFACES ON MAPS/PLANS................................................175LEVELING PROBLEMS.......................................................................................182
V
CHAPTER I
1.1. TOPOGRAPHY – THE SCIENCE OF TERRESTRIAL
MEASUREMENTS
1.1.a. THE ACTIVITY SPHERE OF TERRESTRIAL MEASUREMENTS
The assembly of sciences that contribute to the measurement and representation
of terrestrial surfaces establishes the science of terrestrial measurements. There can be
distinguished three main goals of this science, from the following perspectives:
- Scientific: knowing the shape and dimensions of the Earth, as a planet;
- Direct practical: obtaining topographic plans and maps;
- Indirect-applicative practical: placing, directing and tracing the designed
investments in the field, based on and comply with the execution project.
The main branches of terrestrial measurements (Schema no.1) are:
Geodesy: deals with studying the shape and dimensions of the Earth, or of some
parts of it and with accurately determining the position of some points in the field, which,
as a whole, form the geodetic control network. Because the surfaces that are operated on
are large, the geodetic measurements take into account the terrestrial curvature.
Topography: determines the position in the field of the natural and artificial
details of the Earth’s surface, based on the points of the geodetic network, without taking
into consideration the terrestrial curvature.
Photogrammetry: by processing photographs (photograms) of the terrain, taken
from plane or on the ground, it drafts plans and maps.
Remote sensing: a set of techniques and technologies that allow the remote
analysis of terrestrial surfaces, soil – subsoil, from the qualitative and positional point of
view, by processing the images taken in different regions of the electromagnetic
spectrum.
Cartography: studies the possibilities of passing from terrestrial surfaces – which
are curved, to projection ones – which are plan, scaling down the obtained images and
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representing them on maps, as well as the techniques of drafting, reproducing, printing,
multiplying and depositing topographic maps.
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1.1.b. THE OBJECT AND THE TOPOGRAPHIC APPLICATIONS IN
CONSTRUCTIONS AND MINING
Depending on the solved problem, there exist two components of
TOPOGRAPHY:
GENERAL TOPOGRAPHY, which comprises:
- The study of general methods and instruments, used for different works;
- Measuring and representing terrestrial surfaces of limited extent on
topographic plans and maps (the direct problem of topography).
APPLIED TOPOGRAPHY (or engineering), which consists of:
- Ensuring maps, plans, profiles, bearing points, measurements and
computations (that belong to the direct problem) for the design of different
investments;
- Office and field works for applying the engineering projects and monitoring
the time behavior of the terrains and constructions (the inverse problem of
topography).
General topography, as office science, precedes engineering topography. If the
former has a universally valid character, the latter is profiled and adapted to the
conditions and the domain that it is applied to.
There are many applications of topography in different branches of economy
(Schema 2). But we shall not discuss except those that are directly connected to the
mining domain.
Thus, in constructions, topography precedes, accompanies and follows the
execution works, as we shall see:
- It offers graphical and numerical documentation (maps, plans, known
coordinates benchmarks), which are necessary to study the design alternatives:
- In the phase of technical-economical studies, as well as of drafting the
execution project – integrated in the preceding aspect;
- The designed construction objects, as well as each composing element, are
placed in the field in accordance to the project using topographic means. This
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kind of topographic operations are called tracing and refer to the
accompanying aspect of execution works;
Monitoring the behavior of the foundation terrain and of the construction
elements during the execution is integrated in this category of topographic woks, too.
After finishing the execution of the designed objectives, the above-mentioned
activity is continued until it is found that the deformations in plan (horizontal
displacements) and space (settling) have ceased. These topographic works are integrated
in the following aspect of execution works.
In mining topography also takes part in all phases of the activity: investigation,
design, exploitation, monitoring.
Investigation, phase of mining similar to that of technical-economical studies
from constructions, is solved also through the contribution of topographic methods,
which, besides the maps and plans of the studied area, based on geological laws,
determines the position, shape and dimensions of the ore bodies that can be found in the
terrestrial crust.
In the opening and exploitation activity – similar to the execution in the
construction domain, the mining topography methods contribute to the good progress of
the production processes. The main topographic operations in this study are:
- Topographic surveys of the mining perimeter;
- The exploitation of the opening works;
- Surveys aiming the spatial position of constructions and mining works, and
their support with respect to the ore deposit;
- The correct placement of mining works;
- Tracing works under execution;
- Placing and verifying the position of important mechanical installations.
As the process of exploitation of the ore deposit is carried on, the pressures in the
mining works and the influence of the spaces exploited underground upon the main
mining works and upon the surface are determined based on topographic measurements.
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5
1.2. THE PRINCIPLES OF THE TECHNICAL SCIENCE IN
TOPOGRAPHY
The importance of topography as applied science is undeniable. All this, in the
case in which the execution precision of topographic works is respected and is correlated
to that of the works they are applied on.
In the same way, the leading role of topography in different application domains
should not be neglected, since it implies great responsibility. In order to correspond to
these requirements, the topographic works should be executed respecting the
technological discipline, concisely reflected by the following principles:
1. VERIFYING THE OPERATION:
At least one verification is needed for every topographic work.
2. VERIFYING THE MEASURED DATA:
When the operations in field are finished, the data taken during that measurement
cycle will be verified.
3. THE NECESSARY PRECISION:
The precision of the topographic tracing or measurement works will be given by
the execution precision of the designed objective.
4. APPLYING AUTOMATED CALCULUS:
Data processing is performed, if it is possible, by using means of automated
calculus.
5. THE PERIODIC VERIFICATION OF INSTRUMENTS:
In order to maintain over time the functional qualities of the topographic
instruments (especially the optical ones), their periodic verification and rectification is
required.
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6. FAVORABLE METEOROLOGICAL AND NATURAL CONDITIONS:
There will be performed no work in the field, except if the meteorological and
natural conditions are favorable to the chosen methods and devices. In case of
emergencies, there will be taken such operation measures that the influence of the
environment to be minimal.
7. THE PROFITABLENESS OF TOPOGRAPHIC WORKS:
The choice of methods and instruments used in a topographic operation should
depend on the necessary working precision.
8. GEOMETRIZING THE MEASURED AREA:
The terrain cannot be measured as it is, so it is geometrized. In the choice of the
points by which topographic surface is geometrized, it is essential that the scaled down
image (the plan, the map) obtained as final product to be complete, corresponding to the
requirements of the beneficiary, but not to contain more elements than necessary.
9. AVOINDING LAUNCING INTO THE WORK:
Before beginning a topographic work, there should be drafted a rational activity
schedule, which should be respected along the entire period of execution of the work.
10. RESPECTING THE SAFETY MEASURES OF THE WORK:
In order to avoid any possibility of accident or sickness, the safety measures of
the topographic work and those specific to the domain that is operated within (mine,
construction site, etc.) should be respected. One should work only being completely
healthy.
Schema no. 3 synthetically presents the main measurement and tracing
topographic operations. As it can be seen, two types of angles are used: horizontal and
vertical, and two distances: horizontal and vertical (heights). A clear distinction should be
made between the measurement and tracing operations. In the first case, the linear or
angular ratio under which a series of points existing in the field is to be found is recorded,
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whereas in the second case, one or more dimensional measures are applied in the field, in
order to obtain a new topographic point.
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1.3. LENGTH AND SURFACE MEASURING UNITS
1.3.1. LENGTH-MEASURING UNITS
Along time, there were several measuring units.
In 1790, the geodesists Delambre and Mechein, delegated by the French
Academy, have measured the Earth meridian between Dunkerque and Barcelona, and in
1799, they have proposed the meter as measuring unit in France, which was considered to
be the 1/40.000.000 part of the length of the Earth meridian.
In 1801, there was built a length, called the “standard meter”, which is kept at
Sevres, near Paris. In 1840, the meter was introduced in France, as being mandatory, and
then it was adopted by other countries, too. In our country, the meter was introduced in
1866 by the prince Al.I.Cuza, in order to unify the measurements, which were performed
until then with different length-measuring units.
The last countries that adopted the meter were England and USA, which, until
1971 and 1972, respectively, have used their own length-measuring units.
After some more precise calculus, there was observed that the “standard meter”
represents actually the 1/40,000,003.42 part of the meridian, and because of that diverse
solutions were searched in order to find more rigorous and more stable definitions. Thus,
in 1961, at the General Conference of Measures and Weights, the “standard meter” was
defined to be equal to 1,650,763.73 wavelengths of the orange radiation emitted in
vacuum by the radioactive gas Krypton 84. The multiples and submultiples of the meter
are:
1 m = 10 dm = 100 cm = 1000 mm;
1 km = 1000 m = 10 hm = 100 dam.
In our country, most of the old measurements were performed in stanjeni
(fathoms) or other measuring units. Thus, there can be identified:
stanjenul ardelenesc (Transylvanian fathom):
1 stj = 1.98648384 m or 1 m = 0.5272916 stj;
stanjenul muntenesc (Wallachian fathom):
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1 stj = 1.9666500 m or 1 m = 0.508518 stj;
stânjenul moldovenesc (Moldavian fathom):
1 stj = 2.230000 m or 1 m = 0.448430 stj;
1 prăjină muntenească (Wallachian rod) = 3 stânjeni munteneşti (Wallachian
fathom) = 5.899500 m;
1 prăjină moldovenească (Moldavian rod) = 3 prăjini moldoveneşti (Moldavian
fathom) = 6.690000 m;
1 palmă muntenească (Wallachian palm) = 0.25 m;
1 palmă moldovenească (Moldavian palm) = 0.28 m;
1 dejet muntenesc (Wallachian inch) = 0.02 m;
1 dejet moldovenesc (Moldavian inch) = 0.03 m;
1 linie muntenească (Wallachian line) = 0.002 m;
1 linie moldovenească (Moldavian line) = 0.003 m;
1 (international) marine mile = 1852.20 m;
1 (international) geographic mile = 7420.44 m.
From among the foreign measuring units more frequently used, we can specify:
1 arsin = 0.7112 m;
1 sajau = 2.134 m = 7 feet;
1 veceta = 1066.780 m = 500 sajene;
1 Austrian mile = 7595.94 m;
1 Hungarian mile = 8353.60 m;
1 English mile = 1609.33 m;
1 marine mile = 1852.20 m = 10 cabeltown;
1 geographic mile = 7420.44 m;
1 yard = 0.9144 m = 3 feet = 36 inches;
1 inch (tol) = 0.0254 m;
1 foot (picior) = 0.3040 m = 12 inches.
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1.3.2. SURFACE-MEASURING UNITS
The most known surface-measuring units are those that derive from the metric
system, as follows:
1m² = 100 dm² = 10,000 cm² = 1,000,000 mm²;
1 dm² = 100 cm²;
1cm² = 100 mm²;
1 hectare (ha) = 100 ari = 10,000 m²;
1 ar = 100 m²;
1 km² = 100 ha.
The most important old Romanian surface-measuring units, expressed in square
meters, are:
stânjenul pătrat ardelenesc 1 stj² = 3.59565055 m²;
(Transylvanian square fathom) 1 m² = 0.27803643 stj²;
stânjenul pătrat muntenesc 1 stj² = 3.8671212 m²;
(Wallachian square fathom) 1 m² = 0.2585902 stj²;
stânjenul pătrat moldovenesc 1 stj² = 4.9729000 m²;
(Moldavian square fathom) 1 m² = 0.2010899 stj².
1 prăjină pogonească (yoke pole) = 208.824 m² = 6 prăjini pătrate munteneşti
(Wallachian square poles);
1 prăjină fălcească = 173.024 m² = 4 prăjini pătrate moldoveneşti (Moldavian
square poles);
1 pogon (yoke) = 5011.790 m² (Wallachia);
1 fălcea = 14322.000 m² = 80 x 4 prăjini moldoveneşţi (Moldavian poles);
1 jugăr cadastral (cadastral yoke) = 5754.848 m²;
1 jugăr ardelenesc (Transylvanian yoke) = 5775.000 m²;
1 acru (acre) = 4046.856 m². (See Appendix 1 a and Appendix 1 b)
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Tolerances when measuring and tracing topographic elements
1. Direct measuring of distances
a. Exactly measured lengths T = (0.030 + 0.002L) m (1)
b. Traversing sides outside towns T = (0.004L + l/7500) m (2)
c. Traversing sides inside towns T = 0.003L m (3)
(2) and (3) are increased with 35% for = 5g 10g (slope angle);
(2) and (3) are increased with 70% for = 10g 15g;
(2) and (3) are increased with 100% for > 15g.
2. Measuring horizontal angles with the theodolite
a. The case of one angle T = ec2 = 1.41 ec (4)
where ec represents the reading approximation of the theodolite
ec = 0.2 cc for Theo 010;
ec = 20 cc for Theo 020;
ec = 2 c for Theo 080, Theo 120.
b. The case of multiple angles (horizon tour)
T = ecn (5)
3. Measuring vertical angles with the theodolite
T = ec2 (6)
4. Measuring altitude differences (geometric leveling)
Leveling of order I T = 0.1 mm;
Leveling of order II T = 0.2 mm;
Leveling of order III T = 0.5 mm; (7)
Leveling of order IV T = 1 mm;
Leveling of order V T = 2 mm;
5. Planimetric traverse
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a. Measuring sides TL = 0.003L (see 1 b, c); (8)
b. Closing discrepancy on orientations T = pn; (9)
n = the number of measured angles (compensated orientations);
p = the precision of the reading device
p = 2 cc for Theo 010;
p = 1 c for Theo 020;
p = 10 c for Theo 080, Theo 120;
c. Closing discrepancy on coordinates
T = 0.003D + D/100; (10)
D is the total length of the traverse.
6. Leveling traverse
Leveling network of order I T = 0.5 mmLkm;
Leveling network of order II T = 5 mmLkm;
Leveling network of order III T = 10 mmLkm; (11)
Leveling network of order IV T = 20 mmLkm;
Leveling network of order V T = 30 mmLkm;
Where Lkm represents the total length of the traverse, expressed in km.
7. The leveling of surfaces, profiles
Determining the height of a point
T = 0.5 mm;
(12)
8. Works on plans and maps
a. Linear graphical precision
P = 0.2 N; (13)
Where N = the denominator of the scale of the plan.
b. Angular graphical precision
FU = 20cc; (14)
FU = 15’;
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9. Tracing simple topographic elements in the field
- Value tolerances similar to those from points [1] [4];
a. - Direct tracing of a distance:
T = 1 cm / 100 m; (15)
- Indirect tracing of a distance:
T = 1 2 cm / 100 m; (16)
b. - Tracing a horizontal or vertical angle:
T = [1cc 1c]; (17)
c. - Tracing a given height:
T = [0.001 1] mm;
(18)
d. - Elevating (descending) a normal with the topographic square:
T = 5’; (19)
10. Tracing construction elements and works
a. Embankment works
- Linear (dimensional) deviations:
TL = 5 cm; (20)
- Deviations from the designed height:
TC = 2cm; (21)
b. Foundations
- Deviations from the transversal or longitudinal axes:
TAX = 1 2 cm; (22)
- Deviations from the designed height:
TC = 0.5 1 cm; (23)
c. Casing – the strength structure:
- Dimensional deviations:
T = 0.5 cm; (24)
- Verticals:
T = 0.2 cm / m height (25)
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d. Stakes
- Deviations from the transversal or longitudinal axes:
TAX = 1 cm; (26)
- Deviation from the designed superior height (or console):
TC = 1 cm; (27)
- Verticality:
TV = 1/1000 H; (28)
H = the height of the stake.
e. Sustaining walls (similar to d)
f. Beams
- Deviations from the designed axis:
TAX = 1 cm; (29)
- Deviations from the designed height:
TC = 1 cm; (30)
g. Floors
- Horizontality:
T0 = 1 cm; (31)
h. Bridge crane rails
- Deviations from the designed opening:
TC = 1 cm; (32)
- Plan winding:
TF = 0.5 ÷ 1 cm; (33)
- The height of the two wires in cross section:
TC = 0.5 cm; (max. 1 cm)
(34)
Remark: All the presented values have a guiding character. Depending on the
importance of the work, the tolerances can have narrower or larger values in comparison
to those presented.
The Anglo-Saxon system of measuring units
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Table Appendix 1 a
Length-measuring unitsUnit Submultiples m equivalent
1 inch - 0.02541 foot 12 inches 0.30481 yard 3 foot 0.9144
1 fathom 2 yards 1.82881 terrestrial mile 1760 yards 1609.341 nautical mile - 1852.20
Surface-measuring unitsUnit Submultiples m² equivalent
1 square inch - 6.4516 cm²1 square foot 144 square inch 9.2903 dm²1 square yard 9 square foot 0.8361 m²
1 acre 4840 square yards 4046.8400 m²1 square mile 640 acres 2.5899 km²
- - -
Old Romanian length and surface measuring units
Table Appendix 1 b
Length-measuring units Surface-measuring unitsUnit m equivalent Unit m² equivalent
1 stânjen ardelenesc(Transylvanian fathom)
1.896483841 stânjen pătrat ardelenesc
(Transylvanian square fathom)3.5966508
1 stânjen moldovenesc(Moldavian fathom)
2.23001 stânjen pătrat moldovenesc(Moldavian square fathom)
4.9729000
1 stânjen muntenesc(Wallachian fathom)
1.96651 stânjen pătrat muntenesc
(Wallachian square fathom)3.8671222
1 palmă moldovenească
(Moldavian palm)0.28
1 prăjină pogonească(yoke pole)
208.8240
1 deget moldovenesc(Moldavian inch)
0.031 prăjină fălcească
179.0240
1 linie moldovenească(Moldavian line)
0.0031 pogon(yoke)
5012.000
1 palmă muntenească(Wallachian palm)
0.251 falcă
14,320.000
1 dejet muntenesc(Wallachian inch)
0.021 jugăr cadastral(cadastral yoke)
5,754.6412
1 linie muntenească(Wallachian line)
0.002
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CHAPTER II – THE SHAPE AND DIMENSIONS OF
EARTH, PROJECTIONS, REPRESENTATIONS
2.1. THE GENERAL SHAPE OF THE EARTH
The various economic activities carried on at the surface of the Earth or
underground require the representation of some larger or smaller parts of the area of our
planet on plans or maps, or even its whole representation.
The general or detailed representation of Earth on plans or maps needs a series of
measuring, data processing and drawing steps, some having a generally valid character,
others with a particular character.
The terrestrial surface being curved, the main difficulties come exactly from
passing from the real image, on a curved surface, to the scaled down image, on a plan
surface. In the study and representation of terrestrial surfaces we can distinguish:
THE TOPOGRAPHIC SURFACE = real, on which the measurements are
performed, and which is represented on maps and plans: geometrically and
17
simplified. (It represents 29% from Stotal of the Earth). IT CANNOT BE
MATHEMATICALIZED.
THE GEOID = the medium level surface of calm seas, assumed to be
continued under the continents. IT CANNOT BE MATHEMATICALIZED.
(It is used for representing the third dimension: Z – the height).
THE REFERENCE ELLIPSOID = the operative geometric shape which is
closest to the real one. (It is used for planimetric surveys on large surfaces).
V: the vertical – normal to the geoid
N: the normal – normal to the ellipsoid.
2.2. THE DIMENSIONS OF THE EARTH
In the mathematical study of the shape and dimensions of our planet, along time,
there were established several variants of the terrestrial ellipsoid, considered as being
optimal (table 1).
Table 1
Author Year
Half axis Flatness
a - b = ------
a
Big a Small b
BESSEL 1841 6,377,397 6,356,079 1 : 299.2
CLERKE 1880 6,378,249 6,356,515 1 : 293.5
HELMERT 1906 6,378,200 6,356,818 1 : 298
HAYFORD 1909 6,378,388 6,356,912 1 : 297
KRAKOVSKI 1940 6,378,245 6,356,863 1 : 298.3
18
2.3. CARTOGRAPHIC PROJECTIONS. OVERVIEW.
CLASSIFICATIONS
The only possibility to create maps on larger or smaller scales is to represent the
curved surface of the Earth on a plan, or firstly on a surface which can be unfurled (cone
or cylinder). The fundamental problem of a projection system is to transform the
geographic coordinates that determine the point on the surface of the reference ellipsoid
into the corresponding coordinates (X, Y) in the system of the projection plan.
The plan representation of the terrestrial surface is performed by a series of
geometrical rules, expressed through mathematical relations and practical suggestions
that form the PROJECTION SYSTEM. The representation of the elements of the
terrestrial surface (angle, surface, length) does not assume their representation at their
real size, or the representation of all of them. One of the classifications of the
cartographic systems is the following:
19
Table 2
CONRRESPONDING
Surfaces are kept similar
By
the
natu
re o
f th
e di
stor
tion
s
CA
RT
OG
RA
PH
IC P
RO
JEC
TIO
NS
By
the
aspe
ct o
f th
e ca
rtog
raph
ic n
etw
ork
Azimuthal
The projection
is performed on a plan
Perspective Polar
Oblique
Equatorial
Orthographic D=Exterior RDStereographic D=R Interior DRCentral D = R
EQUIVALENTSurfaces are kept
undistorted
Non-perspective
PolarOblique Equatorial
Conical PolarEQUIDISTANTSurfaces are kept
undistorted
CylindricalPseudo-conical
Oblique
ARBITRARYEverything appears
distorted
Pseudo-cylindricalPoly-conical
EquatorialD: the distance from the center of the sphere to the viewed pointCircular
Practically, a biunique and bicontinuous functional link is created:
U’ = f1 (U, V), where (U’, V’) is the coordinate system on a surface;
V’ = f2 (U, V) (U, V) is the coordinate system on another surface.
20
The cartographic network:
The main cartographic network – the plan image of the network of meridians
and parallels on the terrestrial surface.
The secondary (auxiliary) cartographic network – the plan image of a curved
network on the terrestrial network, adequately chosen.
2.4. CARTOGRAPHIC PROJECTIONS – GENERAL PRINCIPLES
a. Azimuthal
a: azimuth
z: zenithal distance
M’: the image of M on Q (projection plan)
a, q: polar coordinates in the plan the projection of Cartesian coordinates
x = q cos a
y = q sin a
q = f(z)
21
DISTORTIONS
R1 (on the vertical) = 1/R · dq / dz
R2 (on the almucantar) = 1/R · q / sin z
p (on the areolar) = qdq / R2 sin z dz
22
2.5. PROJECTION SYSTEMS USED ALONG TIME IN OUR
COUNTRY
Table 2’
Projection name
Projection type Reference ellipsoid
Year of adoption
Central projection
point
Origin axial
meridian
Radius of the null
distortion circle
Properties
CASSINI CONVENTIONALCYLINDRICAL
KRASOVSKI 1876÷1893 - 250 -
BONE EQUIVALENT CONICAL
CLARKE 1895 - 230
46’27”.83-
LAMBERT- CHOLESKY
CORRESPONDING CONICAL
1914÷1918 - + 20 dif. - Keeps the angles under
certain limitations
ST
ER
EO
GR
AP
HIC
Tangent planBudapest
CORRESPONDING PERSPECTIVEAZIMUTHAL
BESSEL 1933
=
28g 21
c 38cc
=
51g
232.78 Keeps the angles and
shapes resemblances
Tangent planTg.Mures
Secant plan Brasov
HAYFORD
23
GAUSS.KRÜGER
TRANSVERSAL CYLINDRICAL
KRASOV 1951 - =210:270 The axial meridian is represented
without distortions
STEREO 70SECANT PLAN
CORRESPONDING PERSPECTIVEAZIMUTHAL
KRASOV 1970 = 51g
= 25g- Keeps the
angles and shapes
resemblances
The stereographic projection with secant plan Brasov
- The projection center at NV from Brasov;
- Distortion of lengths – around 40cm/km;
- C: central point;
The plan image of the circle that passes through the pole of the projection and
the fundamental point is a straight line and it is adopted as 0z axis, and 0x
0y;
The coordinate axes have been translated with 500km towards S-V by
convention, in order to make these coordinates positive;
In order to pass from stereographic lengths or coordinates in tangent plan to
those in secant plan Brasov, there was established a coefficient equal to
24
0.000666667, which determines a distortion of – 33 cm/km in the center of the
projection and of + 65 cm/km at the periphery of the country;
The map sheets: are divided in rectangular shapes, having 60 x 80 cm;
Distortions: 3-4 times smaller than in GAUSS;
Advantages: only one system of coordinates is carried out on the whole
territory of the country, so there is not necessary, as in GAUSS, to transform
the coordinates from one meantime zone to another;
The projection areas do not have to be limited anymore.
l, l’: the lengths on the ellipsoid.
lt, l’t: the lengths projected on the tangent plan.
ls, l’s: the lengths projected on the secant plan.
25
CHAPTER III – TOPOGRAPHIC ELEMENTS OF THE
TERRAIN
3.1. TOPOGRAPHIC SURFACE, DETAILS, GEOMETRIZING THE
TERRAIN, CHARACTERISTIC POINTS
Measuring and representing on large scale (1:5000 1:200) some terrain surfaces,
which we shall call TOPOGRAPHIC SURFACES, is necessary for different purposes,
especially for designing investments.
The topographic surfaces contain several natural and artificial elements, which are
interesting or not from topographical point of view. We call the topographically
measurable elements of the terrain DETAILS.
Details can be:
NATURAL DETAILS: relief elements, waters; we include here also the
destination of the terrain: forest, vineyard, orchard, agricultural terrain, etc.
ARTIFICIAL DETAILS: diverse constructions, communication means, and
artworks, hydrotechnical works, various networks, etc.
Details cannot be measured on the whole, therefore, for topographic purposes
they are replaced by CHARACTERISTIC POINTS.
The CHARACTERISTIC POINTS are the minimum number of points correctly
chosen to represent the measured detail, on the required scale and detail degree.
GEOMETRIZING THE TERRAIN represents replacing a topographic surface by
the interesting details, and then, by characteristic points, for the purpose of topographic
survey.
The characteristic points are chosen in the points where the detail contour changes
direction and in the points of declivity change.
If the distance between the characteristic points is large (> 50 m), then
intermediary points are chosen on the contour of the detail, which will be called
THICKENING POINTS.
26
PROJECTIONS, MAPS, PLANS
In geodetic measurements the curvature of the Earth is taken into account,
because the measured surfaces are large. The points measured on the real surface of the
EARTH are afterwards projected on the terrestrial ellipsoid, operation that is called
GEODETIC PROJECTION. It can be seen that the projection lines converge towards the
center of the terrestrial reference ellipsoid.
The CARTOGRAPHIC PROJECTION is the operation by which a plan image is
given (through mathematical transformation relations) to the curved image from the
ellipsoid, using a horizontal projection plan. This image, scaled down and
cartographically processed, represents the TOPOGRAPHIC MAP.
The surfaces measured in topography are small, so that the terrestrial curvature
can be ignored, and the projection of the measured points is performed directly on a
horizontal projection plan. The operation carried on through verticals is called
TOPOGRAPHIC PROJECTION.
27
The obtained image, scaled down and topographically processed, represents the
TOPOGRAPHIC PLAN.
The points M, N, P, R represent measured points, that is characteristic points and
points of the control network in the measurement.
TOPOGRAPHIC ELEMENTS OF THE TERRAIN
They determine the relative position of characteristic points in the space.
28
The TOPOGRAPHIC ELEMENTS can be:
- LINEAR: the horizontal distance Diy, slanted distances Lij, absolute heights
Zi, Zy, relative heights – altitude differences Ziy (the last two are vertical
distances);
- ANGULAR: horizontal angles i, vertical angles Vij, declivity angles ij (the
last two are vertical angles).
29
a) Vertical section through the AB
alignment
A, B are two topographic (control or
characteristic) points (de sprijin sau
caracteristice) from the terrain.
b) Two alingments intersected in A
(VB), (VC) vertical plans through A,B,
and A, C, respectively.
(HA) horizontal projection plan through
the point A
ORIENTATIONS, COORDINATES
DIRECTIONS, HORIZONTAL ANGLE, VERTICAL ANGLE
In the horizontal plan, using a graduated circle (the horizontal circle of the
theodolite), placed in its center it coincides with the topographic point A, the axes that
unite the stationed point A with the aimed points B are called ORIENTED
30
DIRECTIONS. Taking into account the sense of the graduations of the circle, it will
result that the horizontal angle will be:
= direction C – direction B.
In vertical plan, using a graduated circle (the vertical circle of the theodolite)
placed in the point A, the axis that unites the point A with the point B is called
SLANTED DIRECTION and it expresses the value of the ZENITHAL ANGLE VAB. It
can be seen that the activity (vertical) angle AB will be:
AB = 100 g – VAB.
In fact, because the device cannot be placed at the level of the stationed
benchmark, the axis AB, and AC, respectively, will be translated with a height
corresponding to the height i of the device with which the point A was stationed.
COORDINATES AXES, ORIENTATIONS
A rectangular systems X0Y is used in topography for repeating the measured
points on the topographic plan, which is chosen such that the 0X axis to be parallel to the
NORTH direction.
In this system, the measured points will be characterized by the values (Xi, Yi)
called ABSOLUTE COORDINATES, in this case, for A: (XA, YA), and for B: (XB,
YB). There also can be identified: (XAB, YAB) called RELATIVE COORDINATES,
noticing that: XAB = XB – XA, YAB = YB – YA and that DAB = X2AB + Y2AB.
ORIENTING A DIRECTION represents the angle measured in right-handed
direction, from the NORTH direction towards that direction.
For each point from the terrain there can be defined three NORTH directions:
Ng = the direction towards the GEOGRAPHIC NORTH;
Nm = the direction towards the MAGNETIC NORTH;
N = the TOPOGRAPHIC NORTH, the direction that is parallel to the 0X axis.
: magnetic declination angle, continuous variable;
: meridians convergence angle (in everyday practice, it is pursued that 0);
The MEASURE of an angle can be any value between 0g – 400g.
31
In figure 9 it can be seen that:
0 < AB < 100g therefore it belongs to the quadrant I;
100g < AC < 200g - “ - II;
32
200g < AD < 300g - “ - III;
300g < AE < 400g - “ - IV.
THE TRIGONOMETRIC CIRCLE, THE TOPOGRAPHIC CIRCLE
In topography, the trigonometric circle was modified as follows:
- The 0X axis became vertical axis, parallel to the NORTH direction;
- The graduation of the circle is in the centesimal system;
- The graduation sense: right-handed direction;
- The angles defined in the circle are codified with the Greek letter (THETA),
having the end points of that direction as indexes – example: AB.
THE RELATION BETWEEN COORDINATES AND ORIENTATIONS
In everyday practice there can appear two cases in what concerns the relation
between the known elements and the required ones (orientations, coordinates).NX
XB
XA
0
Figure 3.12. Orientations and coordinates
A
YA YB
BAB
XAB
YAB
33
CASE I:
A: topographic benchmark, point designated in the terrain.
(XA, YA) : known elements.
(DAB, AB) : measured elements (therefore known).
B: point in the terrain, which can be a new topographic benchmark or
characteristic point.
(XB, YB): required elements.
Computations: XAB = DAB cos AB
YAB = DAB sin AB
XB = XA + XAB
YB = YA + YAB
CASE II:
A, B: some points in the terrain (benchmarks, characteristic points)
(XA, YA), (XB, YB): known elements; (DAB, AB): required elements.
Computations: 2AB
2ABAB sYXD
AB
ABAB X
Ytg
THE CORRESPONDANCE OF THE FUNCTIONS IN THE FOUR
QUADRANTS
34
Trigonometric
functions
Quadrant I Quadrant II Quadrant III Quadrant IV
1 = 1 2 = 2 –100g 3 = 3-200g 4 = 4-300g
sin iy + sin 1 + cos 2 - sin 3 - cos 4
cos iy + cos 1 - sin 2 - cos 3 + sin 4
tg iy + tg 1 - ctg 2 + tg 3 - ctg 4
ctg iy + ctg 1 - tg 2 + ctg 3 - tg 4
Orientation
ij
Xij Yij Orientation
ij
Computation
relation
Example
figure (9)
Quadrant I + + Quadrant I
iy
iyiy X
Yarctg
AB
Quadrant II - + Quadrant II
iy
iygiy sX
Yarctg100
AC
Quadrant III - - Quadrant III
iy
iygiy X
Yarctg200
AE
Quadrant IV + - Quadrant IV
iy
iygiy sX
Yarctg200
AD
The tables complement the knowledge needed for solving the two problems,
regardless of the quadrant in which is the orientation iy. The handbook of tutorials and
problems gives different computation examples, numerically extending the solution of
the two cases discussed earlier.
It should be noticed that from the three tables it results the analysis of the four
previous figures.
PLANIMETRIC AND LEVELING TOPOGRAPHIC SURVEYS,
INTRODUCTORY ELEMENTS
35
The planimetric topographic survey of a terrestrial surface represents all the
operations by which is collected the data needed for drafting the topographic plan of the
measured area, on the scale.
After finding the existence of a sufficient number of control points in the area,
points designated in the terrain with known coordinates (Xi, Yi), the relative position of
each characteristic point (e.g. 1) is measured with respect to a support basis (e.g. 23.22).
This position is given by the following elements: a horizontal angle i (e.g. 1) and a
horizontal distance Diy (e.g. 23.1) obtained from measurements, practically the polar
coordinates of the characteristic point with respect to the support basis.
From figure 3.12 it results the new orientation:
23.1 = 23.22 + 1 (-400g)
Remark: if summing up the known orientation with the horizontal angle it
exceeds 400g, than subtract those 400g from the obtained value. Then, applying the
computational model from CASE I (the relation between coordinates and orientations)
the absolute coordinates of the surveyed point are obtained.
The problem can be extrapolated to any necessary number of measured
characteristic points, solving in this way, from the main point of view, the problem of
planimetric survey of the area that was operated within.
The leveling topographic survey of a terrestrial surface represents all the
operations by which is collected the data needed for completing the planimetric
topographic plan made in the previous stage with data concerning the heights of the
characteristic points from the area.
Remark: in everyday practice, these two operations of PLANIMETRY and
LEVELING are executed simultaneously, collecting the data needed for computing the
complete position (Xi,Yi, Zi) of the measured characteristic point.
After finding the existence of a sufficient number of leveling control points in the
area, points designated in the terrain with known height (e.g. point 37), the data needed
for measuring (or computing) the altitude difference between the two points are collected
(e.g. Z 37.1), obtaining the height of the measured characteristic point from the next
relation:
Z1 = Z37 + Z37.1
36
The elements needed for computing the heights of all characteristic points situated
within an area can be measured with respect to a benchmark of known height found in
that area, solving in this way, from the main point of view, the problem of leveling
survey of the area that was operated within.
37
CHAPTER IV – ERROR ANALYSIS IN TERRESTRIAL
MEASUREMENTS
4.1. MEASUREMENT CLASSIFICATION
The topographic measurements of distances and angles, from the point of view of
the relations created among the measured elements or among them and other elements
obtained by data collecting, can be:
DIRECT MEASUREMENTS: when the value of the measured elements is
obtained by comparing it to a standard (e.g.: the distance measured with a measuring
reel);
INDIRECT MEASUREMENTS: when the value of the determined elements is
obtained by processing some measured data (e.g.: the horizontal distance D ij, obtained
from the relation: Dij = Lijcosiy, where Lij and ij have been measured directly);
CONDITIONED MEASUREMENTS: when direct measurements are constrained
through certain conditioning relations (e.g.: the sum of the measured angles around a
point must be 400g).
The DIRECT, INDIRECT or CONDITIONED MEASUREMENTS, depending
on the operation circumstances under which they were performed, can be:
MEASUREMENTS WITH THE SAME PRECISION: when the measurements
are performed under similar circumstances (instrument, environment, operator), fact that
confers equal confidence to all measurements;
MEASUREMENTS WITH DIFFERENT PRECISION: when the measurements
are performed with instruments in different environmental conditions, fact that can lead
to a greater confidence degree of some measurements in comparison with others.
THE VALUES OF MEASUREMENTS
The results of measurements are called VALUES.
Values can be:
38
REAL VALUES (Xi): value that cannot be obtained, since it is a theoretical,
reference value (which one tends to).
MEASURED VALUES (Mi): the result obtained by measuring a variable, under
accepted measuring circumstances; in practice the variable is measured many times (e.g.:
with the same precision), so individual values Mi are obtained, which slightly differ
among each other, therefore they contain errors.
From probability calculus it can be proved that the arithmetic mean M of these
individual values Mi (in the case of measurements with the same precision) or the
weighted arithmetic mean M0 of these individual values Mj, with weights pj, represents
the value that is closest to the real one. These values are called:
MOST LIKELY VALUES (M or M0) and are computed, according to GAUSS
notation, as follows:
(4.1)
(4.2)
where pj represents the weight coefficients distributed to each individual
measurement.
Remark: The next assertions result naturally:
- The more precise the used instruments are, and the more experienced the
operators are, operating in known environmental conditions favorable to the
measurements, the better the results shall be;
- The greater the number of measurements for a given variable, the closer the
most likely value M (or M0) shall be to the real value;
- The real value being an ideal variable (therefore unknown), it is replaced in
the computations by the value M (or M0).
39
4.2. NOTIONS CONCERNING ERRORS
The large differences between the measured values Mi and the reference value X
(we accept that it is replaced by M or M0) are called MISTAKES. The values incorrectly
measured are removed from the data processing, being unacceptable.
The small (acceptable) differences between the measured values Mi, obtained at
each measurement of a variable, and the reference value X (M, and M0 respectively) are
called ERRORS.
The (INHERENT) appearance of errors is determined by three main causes:
- Equipment causes (resulting from their construction or exploitation), which
determine EQUIPMENT ERRORS;
- Human causes (lack of experience, the limit of sense – especially the visual-
optical one, tired operator), which determine PERSONAL ERRORS;
- Natural causes (different meteorological conditions more or less adequate for
topographic measurements), which determine EXTERNAL
ENVIRONMENTAL ERRORS.
The difference between some two values Mk, and Mp from among the series of
individual measurements executed upon the same variable, is called DISCREPANCY
():
(4.3)
The MAXIMAL DISCREPANCY (max) represents the difference between the
greatest value and the smallest value from among the measurement series:
(4.4)
The TOLERANCE (T) is the maximal admissible discrepancy.
The ACTUAL ERROR (ei = Mi – X) is an unknown value, therefore it is not used.
The PERCEPTIBLE ERROR (Vi = Mi – M) is a value that replaces the actual
error in studies, taking the means M and M0 as reference values.
The ROUGH ERRORS (MISTAKES) are those errors that exceed the tolerance:
e > T or max > T
If in the series of measured values there exists one or more such values that fit
into this class, these are removed from the computation.
40
PROPER ERRORS are those errors that fulfill the condition:
E T or max T (4.5)
PROPER ERRORS can be classified depending on their way of action into:
SYSTEMATIC ERRORS: determined by permanent causes, maintaining the
magnitude and sign, or varying the magnitude by a known law.
The SYSTEMATIC ERRORS:
- Are controllable;
- Can be determined by the influence of the environment, instruments,
measuring methods;
- Are propagated with the number of measurements – therefore becoming
dangerous, since they can alter the final result;
- Must be removed from measurements (improving the measuring
circumstances or applying corrections).
RANDOM ERRORS: determined by unknown causes, expressed as small
variations of different measured values (both as magnitude and sign).
The RANDOM ERRORS:
- Are not controllable;
- Can be determined by the influence of the environment, the performance of
the instruments and of the operator;
- As a whole, they are subject to the probability laws;
- The probability of producing positive and negative errors being the same, the
sum of these errors will be close to zero for the case of a large number of
measurements;
- Small errors are more likely to appear than large ones;
- Cannot be removed from measurements, but they can be diminished –
choosing instruments as reliable as possible, operating under favorable
environmental circumstances, using experienced operators;
The relation: eti = eui n (4.6)
where: eti is the mean total random error;
eui is the mean unitary random error;
n is the number of measurements of the same value;
41
expresses the propagation of random errors.
RANDOM ERRORS in direct measurements
The properties of perceptible errors vi (of random errors) are:
1) [v] = 0 (4.7)
for direct measurements with the same precision,
where:
vi = Mi – M; i = 1,…..,n (4.8)
p · v = 0 (4.9)
for direct measurements with different (weighted) precision.
2) The sum of the squared perceptible errors vi is minimal:
v2 = minim (4.10) and
p · v2 = minim (4.11)
respectively, for the two types of measurements.
The MEAN SQUARE ERROR of one measurement is:
v2 eq = ------ (4.12)
n-1
for the first case, and
p · v2 eq1 = --------- (4.13 )
n-1
for weighted measurements.
eq (and eq0, respectively) characterizes the precision of one measurement.
It was proven that:
Vlim ≤ (2 3) eq (4.14)
or max = Mmax – Mmin ≤ (2 3) eq (4.15)
for the evaluation of some values of the maximal (limit) error and of the
discrepancy max, respectively.
THE MEAN SQUARE ERROR OF THE MEAN will be:
Eq
eM = -------- (4.16) n
42
in the case of direct measurements with the same precision, and
Eq0
eM0 = -------- (4.17) p
This kind of error indicates the closeness degree of the means M and M0,
respectively, to the real value X that they are replacing.
Analyzing the relation (4.17) it can be seen that eM will be smaller if:
Eq is smaller, that is when the work is reliable;
n is greater (optimally, it is recommended that n 5).
RANDOM ERRORS in indirect measurements:
The result y of an indirect measurement can be represented as a function of other
independent variables, directly measured (x1,x2,…,xn), namely:
y = f (x1,x2,…,xn) (4.18)
where xi = the mean values of the independent variables from the direct
measurements,
and if the mean square errors will be denoted with mi, the mean error of the
function f can be computed in the following way:
f ² f ² f ²m2
f = m21 ---- + m2
2 ----- + ……….. + m2n ------ (4.19)
x1 x2 xn
Therefore, it can be said that: THE SQUARED ERROR OF A FUNCTION f IS
EQUAL TO THE SUM OF THE PRODUCTS BETWEEN THE MEAN SQUARE
ERRORS AND THE SQUARED PARTIAL DERIVATIVES OF THE FUNCTION.
4.3. PRESENTING MEASUREMENT RESULTS
The measurement of a variable, once or more times, has a result of general form:
P a (4.20)
where: P is the mean value (M, M0) of the measurement series, after removing the
systematic errors;
a is one of the mean or limit errors (eq, eM, etc.).
43
In the case when the precision of measurements depends on the measured variable
(e.g.: measuring distances), the errors can be expressed as RELATIVE ERRORS (er), for
example:
eM
er = -------- M
Conclusion: ERROR THEORY solves two basic problems in topographic
measurements:
1. It allows removing rough errors (mistakes).
2. It determines the precision of measurements.
The analysis of errors also allows the organization of topographic measurements
(methods, instruments, measuring circumstances, number of measurements), as correctly
and economically as possible.
It should be notices that: ERROR THEORY refers only to
1. Proper errors;
2. Random errors, ONLY AFTER THE MEASUREMENTS HAVE BEEN
CORRECTED OF ALL SYSTEMATIC ERRORS.
CHAPTER V – TOPOGRAPHIC INSTRUMENTS
44
Introductory remark: From the previous chapters it could be seen that
topographic measurements focus on collecting from the field the data needed to compute
the following variables: slanted or horizontal distances, horizontal or vertical angles,
vertical distances – that is, altitude differences. Along time, topographic instruments have
been created and perfected, which are used today to perform measurements with higher
or lower precision, collecting from the field the data needed to compute one or more
variables, even until collecting simultaneously all the data needed to establish the
position of the measured point in the space (complete topographic stations), with manual
or automated data recording or transmitting the data to the center of data processing.
This chapter presents these instruments, their structure and construction, their
usage, and verifying and rectifying these devices.
Initially there are presented the classically constructed instruments, and then the
modern instruments, whose appearance has significantly improved and perfected the
work of the topographer.
5.1. STUDYING THEODOLITES
The THEODOLITE is a device that is used to measure horizontal directions
between two points in the field (a stationed one, e.g. A, and an aimed one, e.g. B or C)
and the declivity angle of these directions with respect to a horizontal plan (generated by
the aiming center of the device Cv).
From the measured directions horizontal angles (e.g. A) and vertical angles (e.g.
AB, AC) are determined.
The theodolites that can measure horizontal distances too, using the optical
method – indirectly, are called TACHEOMETERS.
45
Remarks:
i) There are numerous producers of THEDOLITES-TACHEOMETERS
(Germany, Austria, Switzerland, Hungary, Czech Republic, Sweden, Italy,
Russia, Japan, China, and South Africa), which produce different types of
devices, of different form and precision. Nevertheless, all these devices have
the same main parts and axes;
ii) The theodolites can be classified as:
- Classic theodolites: characterized by the decentralized construction,
with graduated metallic circles, the first ones that have appeared,
46
nowadays being museum artifacts – though they have been produced
until the ’50s;
- Modern theodolites: characterized by the centralized, robust
construction, with graduated glass circles, produced even today, for
more than 40 years;
- Electronic theodolites: mono-block construction, electronic reading,
with the possibility to record the measured variables, produced for
more than 15 years;
iii) Depending on the precision assured for measuring angles, the theodolites can
be classified as:
- Low precision theodolites: equipped with a WIRED reader, the
smallest gradation 10c, the smallest read value 1c, the precision
obtained 2c; for example: THEO 120, THEO 080 – produced until
1990 by Carl Zeiss Jena;
- Medium precision theodolites: equipped with a SCALE reader, the
smallest gradation 1c, the smallest read value 10cc, the obtained
precision 20cc – 30cc; for example: THEO 020, THEO 030 –
produced until 1990 by Carl Zeiss Jena; TT50 MEOPTA – Czech
Republic; TE-D2 MOM – Hungary; Wild T1A, Wild T16 –
Switzerland, etc;
- High precision theodolites: equipped with readers with optical
micrometer, with the smallest gradation 10cc, being capable to read
values of 1cc, the obtained precision 2cc; for example: THEO 010 –
produced until 1990 by Carl Zeiss Jena; wild T2, T3, T4 –
Switzerland; TH2, 3 – Germany.
Specification: until 1990, the main supplier of topo-geodetic equipment for
Romania was Carl Zeiss Jena company (from the former GDR), and now, most of the
devices that exist at the execution structures belong to this category.
MAIN AXES AND PARTS OF A THEODOLITE
47
The device is structured along the following MAIN AXES:
- VV: main axis, vertical during measurements;
- HH: secondary axis, horizontal during measurements;
- 0: reticule – lens, central axis of the telescope;
- NN: the directrix of the level air bubble, tangent axis to the horizontal setting
device of the apparatus.
From the construction of the device:
i) HH VV;
ii) 0 HH;
iii) NN VV;
iv) VV ∩ HH ∩ 0 = {Cv}; Cv: the aiming center.
The device can be rotated around the first two main axes:
R1 rotation around the VV axis;
R2 rotation around the HH axis.
MAIN PARTS:
- Graduated horizontal circle;
48
- Graduated vertical circle;
- Alidade circle, which supports the superstructure of the theodolite and carries
the reading indexes for the horizontal circle;
- The base that supports the entire device;
- The telescope of the device.
The superstructure of the theodolite is the part that has as basis the alidade, being
supported by it: the vertical circle and the telescope.
The infrastructure of the theodolite is the part that connects the superstructure to
the trivet plate, consisting of the horizontal circle and the base.
PARTS THAT ENSURE THE FUNCTIONALITY OF THE DEVICE
PARTS THAT ENSURE THE HORIZONTAL SETTING OF THE
THEODOLITE:
- The level air bubble, the spherical level, the foot screws (three) of the base
(Remark: bubble – similar word level).
PARTS THAT ENSURE LIMITING AND CONTROLING THE MOVEMENTS
OF THE THEODOLITE
- Screw for locking the movement around the VV axis, screw for locking the
movement around the 00 axis, screw for locking the movement around the VV
axis of the horizontal circle (locking the recording movement), device for the
refined movement around the VV axis, device for the refined movement
around the HH axis, device for introducing horizontal angular values, device
that fastens the apparatus to the base.
ACCESSORIES OF THE TELESCOPE THAT ENSURE AIMING AND
POINTING THE MONITORED BENCHMARK:
- Device for focusing the telescope (clarifying the image);
- Device for approximate aiming, screw for clarifying the image of the reticular
plate.
OTHER PARTS:
- The microscope for reading the values of the horizontal and vertical angles,
optical plumb-bob wire = device for optical centering of the device.
49
THE COMPONENTS OF A THEODOLITE
THE TOPOGRAPHIC TELESCOPE
- Is an optical device used for clearly and magnifyingly aiming point (signals);
- Has internal focusing (image clarification) – the reticule is fixed, and the
image is moving in the plan;
- Consists of two coaxial tubes: the lens tube and the ocular tube;
- The lens of the telescope has the purpose to form the image of the aimed
object, reduced, real, reversed (if there isn’t another auxiliary system that
turns the image – upright again), located between the ocular and the center of
the ocular lens;
- The ocular of the telescope has the purpose to magnify the image of the lens;
- The reticule of the telescope consists of a glass plate on which lines are very
finely engraved (1), being called vertical and horizontal cross-hairs, (double
on one side) and stadimetric wires, symmetrically placed with respect to the
previous ones (figure 4).
The technical characteristics of the telescope are:
- Magnifying power, which represents the number that shows how many times
the image of an object seen through the telescope is larger than the image seen
50
with the eye; the value is labeled with M and is given by the ratio between the
focal distance of the lens and of the ocular; practical values of M: 15X60X;
- The aiming field of the telescope represent the conical space bounded by the
generator that passes through the center of the entrance pupil and the interior
border of the bed of the reticular plate; values between 11.5; it is conversely
proportional to its size, the high precision theodolites have large M and a
small aiming field.
THE GRADUATED HORIZONTAL CIRCLE
The graduated horizontal circle (the bearing circle) is concentric with the alidade
circle, having two indexes for reading the horizontal angular values i1 and i2.
- It is fixed during measurements;
- The diameter of the circle is between 70 and 250 mm;
- The smallest gradation can be: 1g, (1/2)g, (1/4)g, (1/5)g, (1/10)g.
The theodolite can be used in two positions, diametrically opposite on the bearing
circle, thus for a measured angle resulting two sensitively equal values:
IA = C I
C - C IB
IIA = C II
C - C IIB
IA + II
A The most likely value will be: A = --------------, only if I
A IIA;
2
Using this method, most of the equipment errors are removed.
The horizontal circle must satisfy the following conditions:
- The graduated circle must be horizontal and stable during measurements;
- The alidade circle must be horizontal and concentric with the graduated circle.
51
THE GRADUATED VERTICAL CIRCLE
The graduated vertical circle (the clinometer) has the purpose to measure vertical
– zenithal angles.
52
- It is assembled in such a way that the line of gradations 0g ………200g is in
the same plan with the aiming axis of the telescope (figure 6);
- It is mobile during measurements, moving together with the telescope;
53
- The reading index J is on the support distaff of the assembly vertical circle –
telescope;
- We will obtain the two vertical zenithal angles VI, VII in the two positions of
the telescope, satisfying the condition:
VI + VII 400g
- The zenithal angle will be:
ZI = VI
ZII = 400g – VII
ZI + ZII
Z = ---------- 2
and the declivity angle of the telescope will be:
= 100g – Z
or, directly from the readings:
I = 100g - VI
II = VII - 300g
I + II
= ----------- 2
THE READING RULE FOR ANGULAR VALUES
THE CIRCULAR VERNIER (Figure 7)
The reading will have two parts:
P I = 261g 30c (because there are three intervals from the gradation 261g to the
origin of the vernier);
P II = 7c (because there are seven intervals on the vernier until a gradation from
the vernier coincides with one on the bearing circle.
54
THE MICROSCOPE WITH LINES (Figure 8)
THE SCALE MICROSCOPE (Figure 9)
Vertical circle:
55
- Exact reading: 87c
- Approximate reading: 80 cc
V = 96g87c80
Similarly, on the horizontal circle Hz = 28g03c60cc
USING THE THEODOLITE
THE PLACEMENT IN THE STATION
Is the operation by which the device is placed in a correct position, ready for
measurements.
The conditions that must be satisfied are the following:
1) It should be placed very stably in the field (the shoes of the trivet should
be thrust all the way into the ground, without forcing);
2) The plate of the trivet should be horizontal;
3) The height of the trivet should allow the operator to perform
measurements in a comfortable manner;
4) The center of the trivet, determined by the center of the plate, should be
above the station point (the point A in this case), on its vertical (VA, VA),
56
which can be verified and accomplished using a plumb-bob wire attached
to the trivet;
5) The theodolite should be stably placed on the plate of the trivet, in a
central position;
6) The main axis of the theodolite should be in vertical position and should
coincide with the vertical of the station point (VV VA VA);
automatically, HH will be placed in a horizontal position, as well as the
horizontal circle and the alidade.
Both the correctness of the measurements and their precision depend first of all on
the INTEGRAL satisfaction of the above-mentioned conditions.
The order of the operations in the field, in order to satisfy these conditions, will
be:
- Verify the station point (whether it was deteriorated or moved);
57
- Open out the legs of the trivet, raise it up (according to condition 3);
- Bring the trivet above the station point, attach the plumb-bob wire and satisfy
simultaneously the conditions 1, 2, and 4;
- Remove the device from its case, verify it;
- Fasten the theodolite on the trivet, temporarily, preliminarily satisfying
condition 6;
- Horizontally set the theodolite using the level air bubble (approximately);
- Successively, horizontally set using the level air bubble – center using the
plumb-bob wire, until condition 6 is completely satisfied;
- Condition 5 is fulfilled without disturbing the position of the device.
The definitive horizontal setting is performed on normal directions (we can guide
ourselves by the axes of the foot screws), watching that the bubble of the level to remain
in central position, in each position rotated around the vertical axis VV of the device.
PERFORMING MEASUREMENTS
From a station performed using the theodolite, aim towards at least other two
points (e.g. B and 1, but could also be 2, 3, etc.).
From among these points, one point is currently another topographic benchmark
(e.g. B), and the other points will become bearing points or are characteristic points of the
details from the area.
Collecting the characteristics of any of these points is similar, therefore we shall
present the steps for measuring the first point (B). These are:
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- Measure the height “i” of the instrument in the station;
- Fix the device in position I (the vertical circle is on the left of the telescope);
- Unlock the rotation movements around the VV and HH axes;
- Approximately aim the signal from the point (B), lock the movements
previously unlocked;
- Focus the image of the signal;
- Using the refined movement screws, bring the aiming center in coincidence
with the mathematical point of the aimed signal;
- Collect the angular values and other data (rod readings, etc.);
- Unlock the device and rotate in right-handed direction towards the second
measured point, which appears first (in this case, point 1);
- Repeat the previous operations.
The measurements can be repeated in position II (the vertical circle is on the right
of the telescope), the rotation sense of the device will be in left-handed direction.
Usually, for both positions of the telescope, the measurements begin and end on
the first aimed point – the known one (B in this case).
During measurements, the following conclusion should be taken into account,
since it derives from the description of the functioning principles of the device: the less
the theodolite is moved or touched, the more precise the collected values will be. For
that:
- Locking and unlocking the device should be performed very delicately;
- The device should not be moved unless necessary;
- Any operation on the device should be performed delicately;
- THE TRIVET SHOULD NOT BE TOUCHED BY HAND during
measurements (this being the most frequent mistake that beginners do).
Attention: collecting data should be performed only from very clear images, both
of the aimed signal and of the readings from the microscope.
ONLY VERIFIED DEVICES WILL BE USED!
VERIFYING AND RECTIFYING THEODOLITES
59
Using devices determines their derangement in time, introducing inadmissible
(ROUGH) errors in performing measurements.
This is why, before usage, they will be verified and rectified PERIODICALLY
(3-6 month).
The construction conditions of the theodolite are:
- The coincidence of the centers of the alidades with the centers of the
graduated circles;
- The normality of the graduated circles on their rotation axes.
Removing errors produced by not satisfying – within acceptable limits – these
conditions is accomplished by averaging the values from the two positions of the
telescope of the theodolite.
The geometric conditions that the theodolite has to satisfy are:
1) The main axis should be vertical (NN VV);
2) The aiming axis should be normal to the secondary axis (0 HH);
3) The secondary axis should be horizontal (HH VV);
4) The line of reading indexes from the vertical circle should be in a horizontal
plan.
Not satisfying these conditions determines adjustment errors, which can be
observed through the checking operations and can be minimized by rectification
operations.
1) (NN VV) ESTABLISHING THE WAY THE CONDITION IS
SATISFIED:
- Verify and rectify the level air bubble;
- Horizontally set the theodolite;
- If rotating the device around the VV axis, the bubble of the level does not
remain in central position, then it means that VV is not normal to the
horizontal circle.
The RECTIFICATION of this derangement is performed only by the producer.
2) (0 HH) is determined by the repositioning of the center of the cross-hairs
from the geometric axis of the telescope, and the rotation axis of the telescope
around the HH axis will describe a CONE, not a vertical plan. This error is
60
called COLLIMATION ERROR (c). ESTABLISHING THE WAY THE
CONDITION IS SATISFIED:
- Install the theodolite in the station and aim a remote point P in position I, read
the horizontal value PHZ1;
- Aim the same point P in position II, reading the horizontal value PHZ2. If PHZ2
= PHZ2 + 200g then there id no collimation error.
- Otherwise, the difference represents the double of the collimation error.
THE RECTIFICATION OF THE ERROR
- Compute the actual reading PHZ2 in position II in which is the telescope:
PHZ2 = 1/2 [(PHZ2 + 200 g) + PHZ2]
which is introduced in the device from the screw of refined movement around the
VV axis;
- It can be seen that the vertical cross-hair has moved from the boundary of the
aimed point P with a distance equal to the collimation error;
- Bring the cross-hair to coincide with the point P, using the horizontal
adjusting screws of the reticule;
- Repeat this operation until the collimation error becomes null;
- Averaging the values obtained in the two positions of the telescope, the
collimation error is eliminated.
3) ERROR OF NON-HORIZONTALNESS OF THE SECONDARY AXIS (HH
is not normal to VV)
ESTABLISHING THE EXISTENCE OF THE ERROR
- Aim a point R locates as high as possible on a vertical wall (figure 12), in the
position I of the telescope, project R in RI by plunging the telescope around
the HH axis, similarly in the position II is obtained RII. If RII RI then the
error exists.
61
- The error cannot be rectified except in specialized shops (CICLOP Bucharest,
IGFCOT Bucharest, DTM Bucharest).
4) THE INDEX ERROR OF THE VERTICAL CIRCLE
ESTABLISHING THE ERROR is performed similarly as the operation from
point 2, except that here the zenithal readings PIV, PII
V are recorded.
- If PIV + PII
V 400g then the error exists;
- Its value will be 2ei = (PIV + PII
V) - 400g;
- The correction through computation is performed computing e i and
subtracting it from the two values PIV, PII
V obtaining the correct values;
- The RECTIFICATION of the error can be done only in specialized shops.
5) SATISFYING THE CONDITION OF CORRECT POSITIONING OF THE
CROSS-HAIRS
ESTABLISHING THE EXISTENCE OF THE ERROR is performed in the shop,
aiming a plumb-bob wire with the telescope of the device; if the vertical cross-hair does
not have the same direction as the plumb-bob wire, then the error exists.
THE RECTIFICATION OF THE ERROR is performed rotating the reticule, after
the screws that were fixing it have been loosened up.
- After rectification, the second condition is verified again.
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Attention: THE VERIFICATIONS IS PERFORMED IN THE ORDER IN
WHICH WERE PRESENTED HERE.
MEASURING ANGLES WITH THE THEODOLITE
THE SIMPLE METHOD (Figures 13, 14, 15)
The method is used when isolated angles are measured.
The measurement is performed in the two positions of the telescope, recording the
readings:
- CIB, ZI
B, CIC, ZI
C readings on the horizontal and vertical circle, telescope in
position I for point B and C, respectively;
- CIIB, ZII
B, CIIC, ZII
C similarly for position II of the telescope.
COMPUTING THE HORIZONTAL ANGLE
IA = CI
C - CIB
IIA = CII
C - CIIB
IA + II
A
A = ------------- 2
COMPUTING THE VERTICAL ANGLE (DECLIVITY OR SLOPE ANGLE
OF THE TELESCOPE) B:
IB = 100g - ZF
B
63
IIB = ZII
B - 300g
IB + II
B
B = ------------ 2
Remark: C is obtained similarly.
For measuring the vertical angle (figure 16) it is taken into account that during
measurements the device will be situated at some height with respect to the stationed
benchmark i, and the signal aimed in the point B will be seen (observable) at some height
s.
If s i then the angle obtained through measurement will be exactly the declivity
angle of the terrain B. If s i (the case when the aim situated at height i is covered by an
obstacle) then the vertical angle that results from the measurement will be different than
the declivity angle of the terrain B. Its computation will imply knowing the horizontal
distance between the station point (A) and the aimed point (B): DAB. In this case the
computation of the angle B is possible:
- From the figure i + h = ZAB + s
h = DAB tgAB
ZAB = DAB tgB
thus i + DAB tgAB = DAB tgB + s
64
DAB tgAB + (i – s)
from where tgB = ------------------------- DAB
From this relation it results that if i = s => tgB= tgAB.
THE METHOD OF SERIES (REITERATIONS, HORIZON TOUR)
This is used in the case of measuring multiple horizontal angles (recording the
declivity angle of the telescope for each direction) from a station point.
The measuring protocol in this case is:
- Stationing (centering, horizontal setting, etc.) in the station benchmark;
- Choose the farthest point as first aim (in the case when the first point is not a
topographic benchmark with which the station point forms the support basis);
- Aim the first point in the position I of the telescope and moving in right-
handed direction aim the other points (e.g. figure 17), the last aim being the
starting point. The readings for the aimed points are obtained: HZIB, VI
B, HZI1,
VI1, HZI
2, VI2, …, HZI
B, VIB, where, as it can be seen, for the first point initial
values are obtained, denoted with ¯ and final values denoted with _ (measured
in right-handed direction);
65
- In the second position (measured in left-handed direction) the following data
is obtained: HZIIB, VII
B, HZII4, VII
4, HZIIB, VII
B, HZII4, VII
4, HZII3, VII
3, …, HZIIB,
VIIB.
It should be mentioned that VIB and VII
B could be neglected, since they have no
relevance in data processing.
Processing measurement data can be performed using a table.
INSTRUMENTS FOR DIRECT MEASUREMENT OF DISTANCES
Depending on the measuring precision, the instruments can be:
- Precise – used for regular topographic measurements: tapes, measuring reels,
and steel wires and their accessories;
- Very precise – used for measuring geodetic bases: invar wire and the
necessary accessories.
The development of electronic instruments for measuring distances, even of
electronic measuring reels, has limited the direct measurement of distances, which is a
difficult procedure, whose precision is conditioned by several factors (atmospheric
conditions, environment, the skills of the operator).
But there exist cases when this method is still used, for example when an
electronic instrument is not available and the sides of a planimetric traverse have to be
measured (thickening of the network of known points in an operation area).
The TAPES – with divisions from dm to dm, marked by a hole (ø 1 mm) in the
axis, at half meter a metallic plate is marked, and the meter and the ends (0 and 50 m) are
marked by metallic plates with stamped values.
- Centimeters and millimeters are measured with a regular graduated bar;
- The ends are equipped with a ring into which the tensioner is introduced
during the measurement;
- Are calibrated at + 20ºC and a tension force of 15 daN (in these circumstances
the tape has nominal length);
- During measurement the following accessories are used:
66
- The steel pickets: metallic rods (~ 30 cm length, ~ 5 mm section) which
mark the ends of the measured panel, 11 pieces fixed on a ring;
- Two tensioners;
- A thermometer;
- A dynamometer, which is used to ensure that the tape is tensioned during
the measurement under a force equal to the standardization one.
The MEASURING REEL:
- Section 0.1 – 0.3 mm x 8-15 mm, lengths 5, 10, 20, 25, 50, 100 m;
- Calibration at + 20ºC and a tension force of 5 daN.
OPERATIONS ON THE DIRECT MEASUREMENT OF DISTANCES
SETTING OUT: it is the procedure through which there is ensured the coaxiality
of the directions by which the measuring reels (the tapes) are stretched during the
measurement, with the direction given by the ends of the measured panel (figure 19).
The procedure can be ensured with a theodolite place at the starting end of the
measurement (e.g. A) or just with the eyes, using some range poles placed at the ends of
the panel (A, B in figure 19) and at the end of the measuring reel stretched for
measurement.
The operator, staying on the measurement direction, 1-2 m behind the point A,
will indicate the direction on which the measurement must be performed (positions 1, 2,
…, etc.) to the operator from the end of the measuring reel (in 1’, 2’, etc.).
67
If l0 is the nominal length of the measuring reel (the tape), which was successively
stretched out along the measured panel for n times, and the length l1 was measured until
the end of the panel on the last measuring reel that was applied, then the measured
distance will be given by the relation:
LAB = n · l0 + l1
CORRECTIONS APPLIED TO LENGTHS MEASURED WITH STEEL TAPES
The precise measurement of distances (support bases, execution works of high
precision investments – creating the control network) implies also the application of some
corrections, due to the fact that the working conditions differ from the conditions in
which the calibration of the measuring instrument was performed.
THE CALIBRATION CORRECTION (Ck)
If: l0 – the nominal length of the tape;
lk – the actual length of the tape, in the moment of the measurement
Then: Ck = lk - l0
The correction for the entire measured length (we agreed to call it LAB) will be:
LAB
CLABk = Ck -------
l0
THE TENSIONING CORRECTION (Ct)
Fr – F0
Cp = ----------- l0
E · A
Where:F0 – the calibration force;
Fr – the force that was applied for tensioning the tape;
E – the elasticity coefficient of steel, e + 2.1 · 106 daN / cm2;
A – the section of the tape (cm2).
If the entire measurement is performed with the same tensioning force F r, then the
total tensioning correction will be:
68
LAB
CLABP = CP -----------
l0
Otherwise, the correction for each tensioning of the tape is computed, cumulating
the obtained values.
THE TEMPERATURE CORRECTION
Ct = T l0
Where: T = Tr - T0
T0 – the calibration temperature (usually 20C)
Tr – the temperature during measurements;
- the thermal dilatation coefficient of steel, = 0,0115 mm / 1C, 1 m.
For the entire measured length the temperature correction will be:
LAB
CLABT = CT --------
l0
THE CORRECTION OF HORIZON REDUCTION C0 (Figure 20)
In the topographic calculus, the horizontal distance (DAB) is used:
DAB = LAB cos = L2AB - Z2
AB
C0 = DAB - LAB
Where: - the declivity angle of the terrain;
ZAB – the altitude difference between B and A.
69
For variable declivity alignments, the alignment is sectioned in panels with
constant declivity (M1, 12, etc.) and each panel is measured (LiJ and i).
N- Compute DiJ = LiJcosi and in the end DMN = DiJ
M
The corrections are applied in the following order:
L’AB = LAB + CLABk
L”AB = L’AB + CLABT
LIIIAB = L”AB + CLAB
P
DAB = LIIIAB cos
It is interesting to compute, for each correction, which are the limit values for
which the application of that correction is not necessary anymore (For example, if for a
measuring reel with l0 = 50 m, the temperature correction is less than 1mm, then it is
obvious that this correction is not necessary anymore. In this case the temperature
interval for which the correction is not applied must be determined. If we assume that C T
= 1 mm, we shall have 1mm = 0,0115 mm/1C, 1m · 50m. (TL - 20C), from where TL =
70
21.7C, therefore the temperature interval in which the correction does not have to be
applied anymore is: 18.3 21.7 C).
THE PRECISION OF DIRECT MEASUREMENT OF DISTANCES
In optimal measuring conditions (clean alignments that allow the correct
stretching of the measuring reel/tape) the measuring precision of a 50m long instrument
can be 0.5 2cm / 100m.
For some length LiJ, the admissible error will be:
LiJ
eL = ± 0.01 -------- = ± 0.01 LiJ (m)100
Remark: in the case of direct measurement of distances, too, repeating the
measurement (for example, forth – from A to B, back – from B to A) and computing the
length as average of the obtained values improves the precision of the measurement.
ELECTRONIC DEVICES FOR MEASURING DISTANCES
Applying the electrooptic or electromagnetic principle, measuring distances with
these devices is done by recording the forth-back time parsed by modulated light, or radio
microwaves, respectively, from the emitting station (located at one end of the measured
panel) to a reflector (located at the other end), and then back to the reception station
(which is the same as the emitting station).
D = ½ v · t
Where v = the propagation speed of waves – the speed of light;
t = the forth and back time.
Practically, modern devices used so far directly display the measured distance.
GEOMETRIC LEVELING DEVICES
THE MEASURING STAFF (STADIA ROD)
71
The measuring staff is a divisional bar placed vertically in the points in which the
altitude difference is determined. The height of the aiming axis of the level telescope is
measured on the measuring staff with respect to the point signaled by the measuring staff.
THE CENTRIMETRIC MEASURING STAFF:
- Made of wood or aluminium, with length of 2, 3, or 4 m, width of 8-12 cm,
thickness of 1.5-2.5 cm;
- Are graduated every cm, from 0.000 m (this end being placed on the signaled
point) to 2.000 m (or 3.000 m or 3.000 m) at the superior end.
Example: figure 22:
S = 2026
M = 1965
J = 1905
The reading on the middle cross-hair of the reticular plate is used in the
computation of the altitude differences, the other two readings having a double
utilization:
- For determining the distance device-measuring staff using the tacheometric
method (see the next chapter);
- For verifying the central reading:
S + JM = -------- 1 mm
2
2026 + 1905In this case: 1965 = ----------------- - 0.5
2
Hence the readings are correct.
72
PRECISION MEASURING STAFF (Figure 23)
These are measuring staffs equipped with an invar strip and vertical setting
devices – spherical levels, graduated, on the invar strip, every half-centimeter; the
accurate reading is performed by centering a division of the measuring staff between the
two convergent cross-hairs (left or/and right) from the reticular plate.
Thus, the reading in the case presented in figure 23 will consist in the rod reading
C = 784.5 and the reading on the drum (e.g. 612), total:
C = 784.5 + 0.612 = 785.112 cm = 7851.12 mm.
LEVELING DEVICES WITH TELESCOPE
73
The main condition these devices must satisfy during measurements is that the
aiming axis (0) should be perfectly horizontal.
These devices are called LEVEL and have the same main axes as the theodolite,
except the HH axis – the device having only one rotation possibility R1 around the VV
axis. The significance of the other axes is the same as in the case of the theodolite.
Comparing to the theodolite, in what concerns the parts, this devices contains as
main parts: the telescope, the base (there also can be a bearing circle and an alidade) with
the accessories needed for operation (screw for locking the movement around the VV
axis, for the refined movement around this axis, spherical level and air-bubble level, the
accessories of the telescope and, in the case of nonflexible devices, device for refined
horizontal setting).
Depending on the method used for ensuring the basic condition (0 should be
perfectly horizontal in the moment of aiming a measuring staff) the devices can be of two
types:
- NONFLEXIBLE LEVEL – for which the refined horizontal setting is
performed for each aimed direction (most known in our country Ni 004 and
Ni 030 CZJena);
- SEMIAUTOMATIC LEVEL – for which the previous operation is
automatically performed, without the intervention of the operator (most
known in our country Ni007 and Ni025 CZJena).
74
VERIFYING AND RECTIFYING LEVELING DEVICES
These are similar operations as in the case of the theodolite, having as main
purpose that the 0 axis to be perfectly horizontal in the moment of measurement.
The order in which the verification-rectification operations are performed is the
following:
1) (NN VV) – as in the case of theodolites;
2) (VsVs || VV) – the axis of the spherical level should be parallel to the VV
rotation axis.
Rectifying this condition: after satisfying condition 1, horizontally set the device
using the air-bubble level, and if the gas bubble of the spherical level is not centered in
the benchmark circle, correct the position of the bubble using the three adjusting screws
of the spherical level, until the condition is satisfied (setting horizontally the air-bubble
level, the spherical level will be horizontally set, too).
3) The level wire from the reticular plate is not horizontal when the device is
horizontally set.
Aim some point (B) (with the horizontally set device) at the boundary of the
visual field. If by moving the telescope of the device through the refined movement
around the VV axis, the point does not stay on the horizontal cross-hair, then ensure the
satisfaction of the condition using the adjusting screws of the reticule.
4) The vertical plan that contains NN || to the vertical plan that contains 0.
ESTABLISHING THE EXISTENCE OF THE ERROR: place the device with
one of the foot screws towards an aimed point (at 20-50 m), horizontally set the device,
read the value M1 at the central hair, shift the device using the left-axis foot screw, rotate
the right-axis foot screw until M1 is read again. If the gas bubble of the air-bubble level
stays coincident, then the error is null, otherwise, by operating the adjusting screws of the
level the condition will be satisfied.
5) (0 || NN) Not satisfying this condition produces the declivity error of the
telescope (figure 25)
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ESTABLISHING THE EXISTENCE OF THE ERROR: is performed through
middle and end geometric leveling.
- For the first station, the declivity angle of the telescope (given by the error 0
|| NN) is , constant, produces a reading error x1, equal on the two measuring
staffs situated at equal distances from the device.
ZAB = a1 – b1 = (a’1 + x1) – (b’1 + x1) = a’1 – b’1
- Through this procedure the error is removed;
- For the second station, the device is placed near one of the points:
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ZAB = a2 – b2 = a’2 + x2 – b2 = (a’2 – b2) + x
It will result that: x = (a’1 – b’1) – (a’2 – b2)
Hence, a2 = a’2 + x = a’2 + a’1 – b’1 – a’2 + b2 = a’1 + b2 – b’1
From S2 bring the level wire in front of the computed reading a2 using the position
adjusting screws of the reticular plate, maintaining the aim towards the point A.
VERIFYING AND RECTIFYING SEMIAUTOMATIC LEVELS (WITH
ADJUSTER)
These devices have no air-bubble level, and therefore the operational limits of the
adjuster must satisfy the following conditions:
1) VsVs || VV
2) The level wire of the reticular plate should be horizontal;
3) 0 should be horizontal.
TACHEOMETRIC DEVICES
These are devices that allow the optical measurement of distances (indirect
methods) and horizontal and vertical angles.
TACHYMETRY WITH VERTICAL MEASURING STAFF
The device placed in the station, will have its aiming center on the vertical of the
station point (CVEVV).
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If: : the distance between the aiming center CV and the lens;
f: focal distance (the distance between the telescope and the center F);
DAB = D’AB + (f + )
D’AB HBut: ------ = --------
f
h: the distance between the stadia hairs;
H: the generator number (the distance between the projections of the
inferior and superior stadia hair on the measuring staff).
f D’AB = ----- H
h fBut f and h are constant, hence K = -----
H
D’AB = KH
K = 100 (more rarely 50 or 200)
It results that D’AB = KH + (f + )
For modern devices f + = 0
Thus DAB = KH
If the telescope is slanted under an angle ’ (figure 28), we can see from the
schema that since the aiming axis is not normal to the measuring staff (MR), the previous
computational principle cannot be applied. In order to be able to apply the previous
relation, a measuring staff (MF) is built in the point M (the projection of the level wire on
the measuring staff), which is normal to the aiming axis – fictitious measuring staff (MF).
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It results that:
L’AB = KH’
But from SS’M:
H’ 2
cos’= ------ H 2
=> H’ = H cos’
L’AB = KH cos’
But DAB = L’AB cos’
In conclusion:
DAB = KH cos2’
The method allows the computation of the altitude difference (ZAB), too, noticing
that:
i + h = ZAB + M
i: the height of the device in the station;
h: the smaller leg of the right-angled triangle with hypotenuse L’AB;
M: the reading at the level wire (central)
ZAB = h + (i – M)
But h = L’AB sin’
h = KH cos’
And replacing h = KH sin’ cos’
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In the end ZAB = KH sin’ cos’ + (i – M);
And, of course ZB = ZA + ZAB
If the aim on the measuring staff (stadia) is performed such that M = i, then we
shall obtain the following relation:
ZAB = KH sin’ cos’
Taking into account that usually K = 100, and the smallest approximate value on
the measuring staff is 1 mm, the precision for determining the distance using this method
is 100-200 mm/100m measured distance, which makes this method useful for
planimetric surveys, but not for measuring support bases.
SELFREDUCING TACHEOMETER WITH DIAGRAM
These are devices that are used for measuring horizontal distances and altitude
differences directly on a specially constructed measuring staff.
From among them, in our country, the best known is DAHLTA 020 (figure 29).
This is a THEO 020 theodolite, which has a glass disk on which the diagram is
traced out. The disk is concentric with the vertical circle and is fixed in the moment of
inclining the telescope under a given angle. The image of the diagram appears in the field
of the telescope, overlapped on the rod image. In the plan of the image the following
curves appear, forming the diagram:
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- The basic zero curve (C0);
- The distance curve (Cd), having the multiplication constant Kd = 100;
- The altitude difference curves (Ch), symmetric pairs with + or – sign,
depending on the declivity angle of the telescope, having the multiplication
constants Kh = ± 10, ± 20, ± 100.
Two short stadia hairs with constant K’d = 200 appear in the upper part of the
image, being used for measuring slanted distances (Fs). If the readings on the DAHLTA
020 measuring staff (figure 30) are ld for horizontal distances and lh for altitude
differences:
DiJ = Kd · ld
ZiJ = Kh · lh
It can be seen that in order to apply the method correctly, the basic curve will
overlap the zero mark of the measuring staff, situated at 1.400 m from the basis of the
measuring staff.
Computing the height of a point (figure 31) results from the following
equivalence:
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h + i = ZA1 + l0
ZA1 = (i – l0) + h
where h is the altitude difference read on the measuring staff.
The height of the measured point will be obtained as:
Z1 = ZA + ZA1
Z1 = ZA + (i – l0) + h
The precision for determining horizontal distances and altitude differences
depends on the constant of the device and the precision of estimation of the value read on
the measuring staff:
- For distances, the precision is ±10 ÷ 20 mm/100 m;
- For altitude differences the precision will be:
< 5 cm for Kh = ±10;
5 cm ÷ 10 cm for Kh = ±20;
10 cm ÷ 20 cm for Kh = ±100.
SELFREDUCING TACHEOMETERS WITH REFRACTION OR DIVORCED
IMAGE
The distance is determined on horizontal graduated measuring staffs, by the
coincidence of a divorced image, split through refraction (figure 32).
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The prism P situated in the plan of the image will split the aimed rod image into
image 1 (of point M), free image, and image 2 (of point M’), imagine deviated.
The deviation angle is constant, determining the proportionality between L’A1
and H (the rod reading).
LA1 = L’A1 + c
where c is the constant of the device;
L’A1 = H ctg
But ctg = 100 and c = 0 due to the construction of the device, and of the
measuring staff, respectively:
LA1 = L’A1 + c = L’A1 = H ctg = 100 H
THE SELFREDUCING REFRACTION TACHEOMETER REDTA 002
It is the most know device of this type in the countries of the former soviet block
(supplied with equipment produced in the former GDR by Carl Zeiss Jena).
The tacheometer REDTA 002 is a theodolite of type THEO 020 having assembled
an optical-mechanical and reducing gear in the front of the telescope, with the use of
which we could measure distances with a precision of 2 cm / 100 m of measured
distance.
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The tacheometer is equipped with an optical micrometer, which consists of a
graduated drum (T) and a rhombic prism (PC) fixed in front of the upper half of the lens
(figure 34). It can be rotated around a vertical axis, with the use of the drum (T),
obtaining:
R0: the direct radius;
RD: the radius deviated by the prism PC by operating the drum T;
RDP: the radius moved with the use of the prism PC by operating the drum T.
The image of the reading microscope for such devices contains the tangent of the
declivity angle of the telescope under the image read on the vertical circle.
We also present the images of the REDTA measuring staff, the visual field of the
REDTA device during measurements and of the graduated drum, in order to explain the
method used for measuring distances and altitude differences.
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The REDTA measuring staff:
1 – centimetric graduated support;
2 – trivet;
3 – REDTA measuring staff;
4 – internal vernier for measuring distances up to 130 m;
5 – external vernier for measuring distances up to 180 m;
6 – benchmarks for parallactic measurement of distances;
7 – collimator;
8 – aiming benchmarks for the zenithal angle.
The order of operations will be the following:
- Install the device in the station, center and horizontally set it, read the height i;
- Install the measuring staff in the aimed point, centering it at the height i of the
tacheometer in the station, brought to horizontal with the spherical level from
the support and normal to the direction between the two points (station point
and aimed point);
- Aim approximately the measuring staff (as in the case of aiming signals with
the THEO 020 CZJena theodolite), clarify the image, aim exactly operating
the refined movement screws;
- Operate the drum of the device, until a gradation from the vernier (the third
one, in this case) coincides with a gradation from the measuring staff.
Perform the readings:
- Horizontal angle, zenithal angle and the tangent of the declivity angle of the
telescope from the reading microscope of the device;
- Read the value H (a division on the measuring staff = 2 m);
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- Find the vernier reading (a division = 20 cm) which coincide with a gradation
on the measuring staff (the third one on the vernier);
- Read the value on the drum.
In this case:
LA1 = the rod reading H +
the vernier reading +
the drum reading.
ZA1
tg = --------- LA1
ZA1 = LA1 tg
where tg is read in the visual field of the microscope of the device.
OPTICAL TELEMETERS
These are the only devices that can be used for the optical measurement of
distances between the station and an aimed point, without requiring beaconing the aimed
point with a measuring staff. The most used devices have an enclosed variable base, the
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aiming being performed by splitting the image of the aimed point (half of the image is
right, half is deviated under a constant angle ). This angle is called parallactic angle ().
From figure 37 it can be seen that if the right image of the point is obtained
through a prism P1, and the deviated image through a (mobile) prism P2, moving this
prism along the external base of the device, the coincidence of the two half images can be
reached. The base is graduated in such a way that reading the distance can be done
directly on it, through a reading microscope assembled on the prism P2.
It can be seen that: LAB = b ctg.
But ctg = 200 (by construction)
LAB = 200 · b
b is the reading on the base of the device.
The device from this category most known in our country is:
THE SELFREDUCING TELEMETER BRT 006
The distance read on the base will directly be the horizontal distance (if the
reducing gear is coupled) or the slanted distance – if we do not couple this device.
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- The base of the device has 30 cm;
- The multiplication constant K = 200;
- The deployment domain: distances up to 60 m, using the device, and up to 180
m, using coincidence marks installed in the aimed point.
The precision of measuring distances can reach values of 6 cm / 100 m.
PARALLACTIC MEASUREMENT OF DISTANCES
Before electronic devices for measuring distances appeared, the direct
measurement of distances (clumsy and time-consuming) could not be matched by indirect
methods, from the precision point of view. And all this because the value of the (inclined
or horizontal) distance was obtained by applying a multiplication factor (K = 100; 200)
on a read gradation (whose least estimated value can be 1 mm).
Parallactic measurement of distances transforms measuring the distance in
measuring a horizontal angle (: the parallactic angle).
The principle (figure 39) consists in placing (centering, horizontally setting) a
theodolite in one of the points (e.g. A) and building a basis normal to the measured
alignment (MN BA), of known (measured) length b.
It results:
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DAB = b ctg
- The angle is obtained from the difference of the directions AN and AM, and
the basis is built equally distant from the central point B – left – right.
It is obvious that the construction precision of the base b can be very large (up to
2mm / 10 m of base), which can lead to a parallactic distance measuring precision of
20mm / 100 m, comparable to that of direct distance measuring.
In order to facilitate the application of the method, a special measuring staff was
built (similar to the REDTA measuring staff, but not graduated), with base b = 2.000 m.
In this case, for an angle measuring error e = 2cc, it corresponds a distance
measuring error eD = 15 mm / 100 m.
There are mentioned extensions of the method for bigger distances between the
ends of the measured panel or other causes that limit the method (figure 40) (e.g. the lack
of visibility on the left of the point B – case b).
TRIGONOMETRIC METHODS FOR MEASURING DISTANCES
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The trigonometric method is an extension of the previous method, having the
advantage that the aimed point does not have to be accessible (figure 41).
In this case, an auxiliary base DAC is built, designating the point C (required to be
the mobile station with the theodolite). The horizontal angles A, B are measured
(ideally B, too, case in which the angles from ABC can be compensated), resulting the
sinus theorem:
DAB DAC DCB
----------- = ----------- = ----------- sinC sinB sinA
sinC
From where DAB = DAC ---------- sinB
where B is measured or computed B = 200 - ( A + C).
INSTRUMENTS AND DEVICES FOR TRANSMITTING POINTS ON THE
VERTICAL
These are instruments for which aiming is performed on the vertical towards
ZENITH (Z), towards NADIR (N) or in both directions. They have a
determining/transmitting precision on the vertical up to 1mm / 100 m.
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CHAPTER VI – PLANIMETRIC SURVEYS
The TOPOGRAPHIC SURVEY is the totality of topographic works performed in
a certain area, having the purpose of compiling the TOPOGRAPHIC PLAN or MAP.
The PLANIMETRIC SURVEY refers to collecting the necessary data from the
field in order to establish the position in the plan (the coordinates Xi, Yi) of the
characteristic points of the (natural or artificial) measured details.
The LEVELING SURVEY has the purpose of emphasizing the third coordinate
of the measured points, the spatial coordinate (Zi), by determining the heights of the
measured points, emphasizing the relief of the measured area.
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For the entire package of obtained data to have a common denominator, all
topographic measurements are performed in a chosen coordinate system:
X0Y for plan coordinates, Z0: an origin for measuring heights, for Romania, since
1970, the zero height of the level of the Black Sea.
0 is chosen in such a way that all coordinates Xi and Yi, to be positive, on the
entire marked territory.
Assuming that the coordinates of the point A are known in this system: (XA, YA,
ZA), the orientation AC towards another point C, where A and C are bearing points in the
chosen coordinate system, that is, points materialized in the field, and that the slanted
distance LAB towards the surveyed point B and the angle B formed by the direction
between the benchmark A and the point C with known direction are measured, the
position in space of the point results studying figure 2:
- THE RELATIVE POSITION WITH RESPECT TO THE BASIS AC:
(B, DAB), where DAB = LAB cos (figure 2.a) (1)
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- THE ABSOLUTE POSITION IN SPACE, WITH RESPECT TO THE X0Y
SYSTEM and THE ORIGIN HEIGHT Z0 will be:
XB = XA + XAB (figure 2.b)
YB = YA + YAB
ZB = ZA + ZAB (figure 2.a)
where XAB = DAB cos AB, where AB = AC + B (figure 2.c)
YAB = DAB sin AB (figure 2.b)
ZAB = LAB sin tg (figure 2.a)
The basic elements in topographic surveys were presented in chapter III: the
topographic elements of the terrain, details, characteristic points, coordinates and
orientations.
The methods used for planimetric survey will be detailed in the sequel.
Remark: in this chapter there will be discussed only the study of the planimetric
position of the measured points, without details concerning their height.
PLANIMETRIC CONTROL NETWORKS
The planimetric control system X0Y must be represented at the level of the terrain
by a geometric network consisting of points designated in the field with coordinates
known in that system. The shape and size of this network depends on:
- The shape and size of the surveyed surface, and its relief;
- The covering degree of the surface with natural and artificial details;
- The scale of the topographic plan compiled in the end.
The planimetric representation of a surveyed surface is UNITARY,
HOMOGENEOUS, CONTINUOUS AND ACCURATE only if adequate measurement
methods are used, based on a correctly performed geometric network.
In order to ensure the UNITY of topographic measurements on the entire national
territory, there was created (in all countries) a STATE GEODETIC CONTROL
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NETWORK, which covers the entire state territory with a control network consisting of
triangles with apexes of known coordinates (TRIANGULATION NETWORK).
There also exist LOCAL CONTROL NETWORKS, on which planimetric
measurements can be supported, the fundamental condition being the existence of the
connection between the LOCAL SYSTEM and the NATIONAL SYSTEM, that is, to
have the possibility to TRANSLATE the local coordinates into the national system.
THE GEODETIC CONTROL NETWORK – THE GEODETIC CONTROL BASIS
It is built based on the following principles:
- The geodetic basis of planimetric surveys consists of:
- The network of triangulation points;
- The network of traverse (polygoniometry) points;
- The cartographic projection applied: STEREOGRAPHIC 1970, secant plan;
- The origin height for LEVELING: THE BLACK SEA “0” HEIGHT,
fundamental benchmark;
- Reference ellipsoid used: KRASOVSKI;
THE STATE GEODETIC TRANGULATION NETWORK
- It is composed of a network of triangles structured on five orders:
- ORDERS I, II, III, IV that represent the SUPERIOR ORDER
TRIANGULATION;
- ORDER V that represents the INFERIOR ORDER TRIANGULATION;
- The basic condition: cover the entire national territory with known points
through the created triangles;
- It is carried on through chains of triangles, along meridians and parallels, at an
average interval of 200 km, with triangle sides lengths of 20-60 km
(GEODETIC TRIANGULATION CHAINS);
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- A basis is established in each intersection of two chains (6-12 km), which is
measured;
- In each intersection of chains, the azimuth of the measured geodetic basis and
the measured geographic coordinates of one of the ends of the basis are
measured astronomically;
- The intermediary areas of the triangle chains are covered with triangles with
sides of 20-60 km, too, the entire created network representing the
GEODETIC TRIANGULATION NETWORK OF ORDER I;
- Step by step, the triangles are thickened (triangle inside triangle) through
points of order:
- II: triangle sides of 10-20 km;
- III: triangle sides of 7-15 km;
- IV: triangle sides of 4-8 km;
- Thickening order V: sides of 1-2 km (1 point / at most 100 ha).
The computation of these points is performed in the following way:
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ORDER I: the network of points is transposed on the ELLIPSOID, computing the
geographic coordinates (, ), the points are transposed through CARTOGRAPHIC
PROJECTION on the projection plan, computing the Cartesian coordinates X and Y.
ORDER II, III, IV: the computation is performed in the projection plan, taking
into account the terrestrial curvature, coordinates X and Y.
ORDER V: directly in the adopted projection plan, coordinates X and Y.
The POLYGONIOMETRY NETWORKS are rigorously measured and computed
networks, which unite the triangulation points.
The SURVEY NETWORK, made through the method of PLANIMETRIC
TRAVERSING (figure 5), is built in the field in order to serve as basis for computing the
details of the terrain.
Depending on the nature of the points that they are based upon, the traverses can
be:
- MAIN TRAVERSES, based upon TRIANGULATION or
POLYGONIOMETRY points (figure 4);
- SECONDARY TRAVERSES, based on a TRIANGULATION or
POLYGONIOMETRY point, and a point from a main traverse, or, completely
based upon points belonging to a main traverse.
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LOCAL CONTROL NETWORKS
These are applied in the case when:
- There is no state triangulation or the number of points is not adequate in the
SURVEYED AREA;
- The SURVEYED AREA is small (S < 100 km²) and the connection to the
national geodetic system is not justifiable.
The local control network will be created in the following way:
- A polygon with visible diagonals is built (12,34);
- A base is measured (34): D34;
- The geographic (astronomic, magnetic) orientation of a diagonal is measured
(12): 12;
- Arbitrary coordinates (X1, Y1) are assigned for point 1, such that the entire
area to have bearing and characteristic points with positive coordinates in the
chosen system (figure 6).
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- All the angles from the formed triangles are measured, correcting the
measurement errors (the sum of the angles in each triangle should be 200g);
- Compute all the orientations of the other sides, starting from 12 (e.g. 14 =
12 + 1), using compensated angles (i, i);
- Compute the other sides of the triangle (D12, D14, etc.) using the sinus
theorem;
- Compute the relative and absolute coordinates of the other points;
For example:
DX12 = D12 cos 12 (2)
DY12 = D12 sin 12
X2 = X1 + X12 (3)
Y2 = Y1 + y12
Beginning with this known polygon, the LOCAL PLANIMETRIC NETWORK
IS BUILT through:
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1) LOCAL TRIANGULATION NETWORKS (figure 7) – are obtained at 2÷3
km;
2) INTERSECTIONS there are obtained points at 0.5÷1km;
treated in the next chapters
3) TRAVERSES there are obtained points at 0.05÷0.20 km.
On the whole, in the end, BEFORE BEGINNING THE PLANIMETRIC
SURVEY OF THE AREA, all these point have to be sufficient to form the bearing
support for measuring EACH CHARACTERISTIC POINT from the surveyed area.
DESIGNATING AND SIGNALING THE POINTS OF THE PLANIMETRIC
CONTROL NETWORK
DESIGNATING – the procedure of materializing the position of the topographic
point in the field (temporarily or permanently).
SIGNALING – the procedure of marking the aimed points (temporarily or
permanently).
DESIGNATING POINTS
TEMPORARILY: - for an interval of a few years (at most 5 years), it is performed
with:
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- Wood stakes (hardwood: beech, oak), 30-50 cm length, 3-5 cm square section,
with a nail hammered at the superior end, in axis, marking the mathematical
point (whose coordinates are computed), the inferior end is sharpened.
- Metallic pegs, 20-30 cm length, 1.5-2.5 cm section, hemispheric upper end
with a chertat sign ( 1 mm) in the axis, which will represent the
mathematical point.
In both cases, the stakes will be beaten in the ground all the way, such that 2-5 cm
will remain at the surface. Attention: the stakes must be fixed vertically in the ground.
PERMANENTLY (MARKING THE POINTS): - marking with a longer usage
period of the point;
- It is performed using concrete (reinforced concrete) boundary marks, shaped
as a truncated pyramid (upper side 10-20 cm, lower side 20-40 cm, height 60-
100 cm);
- Engrave a metallic peg in the axis, with a hemispheric end, similar to the one
presented previously;
- It is recommended that the marking to be performed underground, too, for the
case when the boundary mark from the surface is destroyed, such that there
would exist the possibility to reconstruct the mathematical point on the
surface (figure 8).
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Thus, after digging the marking ditch, the signal from the underground 1
(underground mark) is placed at the bottom, then a signaling layer (ground bricks) 2, then
the ditch is filled up with the soil resulted from the digging, framing the concrete
boundary mark 4, by marking from the exterior.
Remark: For the signal from the surface to be on the same vertical with the
underground mark, an external marking is performed (figure 9), by intersecting the axes
13 and 24, obtaining the position of the mathematical point P (the axis of the boundary
mark, for which the vertical VV – with the plane coordinates Xp and Yp, are defined).
- Protect the benchmark with a filling layer 5.
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SIGNALING POINTS
It is the operation by which aiming points from the station point is allowed,
signaling the vertical VV of the measured topographic point or characteristic point.
Signaling can be:
- Temporary, only during measurements, which is performed with the use of
the wood or metallic peg (square, hexagonal, triangular or circular section,
with 3-5 cm diagonal), 2 m length, painted alternating in white/red, sharpened
at the lower end, in order to allow the correct placement on the measured
point;
- Permanent: with beacons, towers, pillar signals, called GEODETIC
(TOPOGRAPHIC) SIGNALS (figure 10).
Signaling can be:
- Centric: the axis of the signal coincides with the vertical axis of the signaled
geodetic (topographic) point (figure 10 c, d).
- Eccentric: there exists a measured distance e (the eccentricity of the signal)
between the axis of the signal (VsVs) and the vertical axis of the signaled
geodetic (topographic) point;
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- Another element that has to be measured is the height of the signal (H) with
respect to the height of the signaled mathematical point in the field.
In the case of the pillar signal, used in populated centers, the beacons are placed
on the terraces (roofs) of buildings, on concrete pillars, which allow both stationing with
the theodolite (after the signal was removed) and aiming the point by a signal. Therefore,
it is a centric signal.
Also, unstationable points can be used as signals, which will be used only as
direction points: the peaks of the church spires, lightning rods on industrial buildings.
Regardless of the signaling method, the GEODETIC (TOPOGRAPHIC) SIGNAL
must be: visible and firmly fixed in the ground (tree, building).
THE TOPOGRAPHIC DESCRIPTION OF POINTS (THE MARKING FILE OF
THE TOPOGRAPHIC POINT)
It allows the identification of the position of a topographic point in the field, in the
case when one wants to use it in topographic measurements (figure 11).
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The marking file of the point will contain:
- The coordinates (Xi, Yi), eventually (Zi) of the benchmark;
- The description of the used benchmark;
- At least TWO, optimally THREE distances with respect to known objects
from the field (building corners, electricity or telephone poles, duct tops, etc.).
The position of the topographic point can be reconstructed by linearly
intersecting these distances, identifying it in the field.
COMPUTING THE COORDINATES OF CONTROL NETWORKS
THE METHOD OF INTERSECTIONS
DIRECT INTERSECTION
The points A, B, and C are the geodetic (topographic) benchmarks known in the
field. Therefore, there are known the following:
(XA, YA,); (XB, YB); (XC, YC).
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The point P is the new benchmark, therefore there are performed measurements
(the angles i, i) and computations in order to determine its coordinates (Xp, Yp). It can
be seen that from any combination A and B, B and C, C and A, there result the
coordinates of the point P, the computations being similar. Hence, for the first
combination:
- AB is obtained from the coordinates:
YABtgAB = --------------- (4)
XAB
BA = AB + 200g
(5)
AP = AB + 1 (figure 12) (6)
BP = BA - 2 (figure 12) (7)
YAP YP - YAtgAP = --------------- = ------------- (8)
XAP XP - XA
YBP YP - YBtgBP = --------------- = ------------- (9)
XBP XP – XB
(XP - XA) tgAP = YP – YA (+)
(XP - XB) tgBP = YP – YB (-)
XP tgAP - XA tgAP - XP tgBP + XB tgBP = YP – YA - YP + YB
XP (tgAP - tgBP) = YB – YA+ XA tgAP - XB tgBP
YB – YA+ XA tgAP - XB tgBP=> XP = -------------------------------------------------- (10)
tgAP - tgBP
YP = YA + (XP – XA) tgAP or
YP = YB + (XP – XA) tgBP
105
This first alternative, resulting from the combination A and B, can be verified by
the values obtained from the combinations B and C, and C and A.
If the values are close (within the margins), then the most likely value of the
coordinates of the new point will be the arithmetic mean of the values obtained from the
three combinations (separately for XP and YP, respectively).
It should be noticed that this method allows for a first adjustment of the measured
values – since the sum of the angles measured in the points A, B, and C must be equal to
200g. The difference (within acceptable margins) will be equally corrected on the six
angles, satisfying the previous condition.
RESECTION (INDIRECT INTERSECTION, POTHÉNOT PROBLEM, MAP
PROBLEM)
In this case, stationing in the new point P, aim three known points M, N, and R.
Measure the angles formed in P by the directions towards the three known points (,,)
(figure 13).
Write the analytic equations of the three right lines PM, PN, PR:
(YM – YP) = (XM – XP) tgPM
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(YN – YP) = (XN – XP) tgPN
(YR – YP) = (XR – XP) tgRN
Choosing PM as unknown it can be seen that
PN = PM - ( + )
and replacing it in the group of previous equations, we obtain:
1. (YM – YP) = (XM – XP) tgPM;
2. (YN – YP) = (XN – XP)tg PM - ( + );
3. (YR – YP) = (XR – XP) tg (PM - ).
a system with 3 equations and 3 unknowns: Xp, Yp, tgPM.
Choosing tgPM as the first unknown and solving the system:
1. YP = YM + (XP – XM) tgPM;
2. YP = YN + (XP – XN) tg PM - ( + );
3. YP = YR + (XP – XR) tg (PM - ).
or:
1. YP = YM + (XP – XM) tgPM;
tg PM - tg( + )
2. YP = YN + (XP – XN) ---------------------------- (11)
1+ tgPM tg( + )
tg PM - tg3. YP = YR + (XP – XR) -----------------------
1+ tgPM tg
subtracting the equations 2 and 3 from equation 1, we obtain:
tg PM - tg( + )
1) – 2) = (YM + (XM – XP) tgPM = YN + (XP – XN) ------------------------ (12)
1+ tgPM tg( + )
tg PM - tg1) – 3) = (YM + (XM – XP) tgPM = YR + (XP – XR) --------------------
1+ tgPM tg
107
We proceed, trying to eliminate XP:
tg PM - tg( + )
XPtgPM - XMtgPM + YM – YN – XP ---------------------------- +
1+ tgPM tg( + )
tg PM - tg( + )
+ XN ---------------------------- = XPtgPM - XMtgPM + YM - YR -
XMtgPM
tg PM - tg tg PM - tg- XR ----------------------- + --------------------- = 0 (13)
1+ tgPM tg 1+ tgPM tg
tg PM - tg( + )
XPtg PM - ------------------------ = YN – YM + XMtgPM +
1+ tgPM tg( + )
tg PM - tg( + )
+ XN---------------------------- (14)
1+ tgPM tg( + )
tg PM - tg
XPtg PM - -------------------- = YR – YM + XMtgPM +
1+ tgPM tg
tg PM - tg + XR -----------------------
1+ tgPM tg
We divide the two relations:
tg PM - tg( + ) tg PM - tg( + )
XPtg PM - -------------------------- YN – YM + XMtgPM + XN ----------------------------
1+ tgPM tg( + ) 1+ tgPM tg( + )
--------------------------------------------- = ---------------------------------------------------------------- tg PM - tg tg PM - tg
XPtg PM - -------------------- YR – YM + XMtgPM + XR ------------------------
1+ tgPM tg 1+ tgPM tg
and we obtained an equation with only one unknown tgPM.
Denoting X = tgPM, we shall have:
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X- tg( + ) X - tg( + ) X - --------------------- YN – YM + XMX+ XN --------------------
1+ X( + ) 1+ X tg( + )----------------------------- = ------------------------------------------------------- (15) X - tg X - tg X - --------------- YR – YM + XMX + XR ---------------
1+ X tg 1+ X tg
X + X² tg ( + ) – X + tg ( + ) -------------------------------------------
1 + X tg ( + ) ---------------------------------------------- =
X + X² tg + X tg ---------------------------- 1 + X tg
YN – XYN tg ( + ) - YM - XYMtg ( + ) + XXM + X²XMtg ( + ) + XXN - XNtg ( + )
-------------------------------------------------------------------------------------------------------------------- 1 + X tg ( + )
= -------------------------------------------------------------------------------------------------- (16)YR - XYR tg - YM -XYMtg + XXM + X²XMtg + XXR - XRtg --------------------------------------------------------------------------------------
1 + X tg
X² tg ( + ) + tg ( + )--------------------------------- =
X² tg + tg
YN - XYN tg ( + ) - YM - XYMtg ( + ) + XXM + X²XMtg ( + ) + XXN - XNtg ( +
)= ---------------------------------------------------------------------------------------------------------
X²XMtg + XXR - XRtg
Solving the equation we obtain:
(YN – YM) ctg ( + ) + (YM – YR) ctg + XR – XNX = tg PM= -------------------------------------------------------------------- (17)
(XN – XM) ctg ( + ) + (XM – XR) ctg - YR + YN
and we replace it in the corresponding relations, and we obtain (XR, YR).
COMBINED INTERSECTION
109
Combining the previous methods, we obtain another method, in which the
precision of coordinates computation can be improved, because there exists the
possibility to adjust the measured angles (figure 14).
Therefore, the following three conditions must be satisfied:
(1+1) + (2+2) + (3+3) = 200g (18)
(1 + 2 + 3) = 400g (19)
1 + 2 + 1= 200g
2 + 3 + 2 = 200g (20)
1 + 3 + 3 = 200g
Only after the measured angular values have been adjusted such that the
previously mentioned conditions to be satisfied, we can proceed to computations.
The computation of coordinates id performed through DIRECT
INTERSECTION.
110
The purpose if to thicken the control networks (triangulations, polygonations,
intersections), in order to have the necessary number of known points in the field, on
which the planimetric survey of the area to be based on.
CLASSIFICATION OF TRAVERSES
A. TWO ENDS TRAVERSES, which can be:
1. With two ends and two orientations;
2. With two ends and one starting orientation;
3. With two ends and one ending orientation;
4. With two ends and no known orientation.
B. ONE END TRAVERSES, which can be:
5. With one end and one starting orientation;
6. In closed circuit.
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DESIGNING PLANIMETRIC TRAVERSES
The route of planimetric traverses, their shape and type, are chosen on a
topographic plan of the studied area (scale > 1:5000). The following conditions will be
respected for the design:
- The alignments of traverses should be near the details that will be surveyed
and should cover the entire area;
- The traverse points should be situated in stable, non-circulated areas;
- There should be visibility between the neighboring points of the traverse, and
from them towards the details;
- The length of traverse sides should be within the interval 50-200 m, with an
optimum at 100-150 m, and a total length that should not exceed 3000m;
- The traverse sides should be close in length, and the traverse should be as
linear as possible;
- The instruments for measuring angles and distances should be carefully
chosen, and should be verified before usage.
FIELD OPERATIONS
DESIGNATING TRAVERSE POINTS
MAIN TRAVERSES – the ends of the main traverses will be included in the
control network, and therefore, will be designated by concrete boundary marks (on the
ground, under ground), and the signaling will be performed with a butterfly beacon.
SECONDARY TRAVERSES – designating will be performed with wood or
metallic stakes (temporary designating), and the signaling will be performed with pegs.
MEASURING TRAVERSE SIDES
It can be performed directly with the steel tape or electronically.
Directly, there is measured the tilted distance LiJ, which will be reduced at the
horizon with the relation:
DiJ = LiJ cos i
112
Each side will be measured back and forth, the difference between the value LiJ
obtained measuring forth (from the point i towards the point J) and the value LiJ obtained
measuring back (from J towards i) must be less than the margin Ti:
Ti = 0.003L (22)
If this condition is satisfied, the most likely value of the length of the measured
side will be the arithmetic mean of the two values:
LiJ = LiJ + LJi (23)
LiJ will be corrected based on the principle of applying corrections for direct
measuring of distances.
MEASURING ANGLES FORMED BY THE TRAVERSE SIDES
DECLIVITY ANGLES
- Back and forth, position I, position II (figure 16).
The two means obtained forth iJ and back Ji must be close in value, within the
margin ± 1c.
HORIZONTAL ANGLES
- In each traverse point, position I, position II.
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All the horizontal angles are measured on the same side of the traverse, condition
that is satisfied if each angle is measured from the back side, in right handed direction,
towards the front side (figure 17).
Practically, both in the case of declivity angles and in the case of horizontal ones,
the specifications from the chapter “Measuring angles with the theodolite – the case of
one angle” will be respected.
COMPUTATIONAL OPERATIONS
1. TRAVERSE SUPPORTED IN BOTH ENDS
Known elements:
A, B, C, and D topographic benchmarks of given coordinates:
(XA, YA); (XB, YB); (XC, YC); (XD, YD);
1, 2, … new topographic benchmarks.
Unknown elements:
(X1, Y1); (X2, Y2); …
Remark: only two new points were chosen, to prevent the useless increase of
performed computations; in the case when the traverse has more than two new points, the
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computations are the same, adding the computational elements corresponding to the other
points:
Measured elements:
- Horizontal angle i (position I, position II);
- Declivity angle i (position I, position II, back and forth);
- Slanted distances LiJ (back and forth).
COMPUTATIONS
- THE MEAN OF THE MEASURED ELEMENTS
- The average length of the slanted distance LiJ = LiJ + Lji;
iJ + Ji
- The average declivity angle iJ = ---------------- (24) 2
i’ + I”- The average horizontal angle i = --------------
2
In order to simplify the notations, these values will be denoted with (LiJ, iJ, i).
COMPUTING HORIZONTAL DISTANCES AND ALTITUDE DIFFERENCES
DiJ = LiJ cos iJ
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ZiJ = LiJ cos iJ
a) COMPUTING SUPPORT ORIENTATIONS
From the coordinates of the bearing points, analytically results:
YAB YB - YAtg AB = ---------- = ------------ initial support orientation;
XAB XB - XA
(25) YCD YD - YC
tg CD = ---------- = ------------ final support orientation;
XCD XD - XC
b) COMPUTING THE COARSE ORIENTATIONS OF THE TRAVERSE
SIDES (figure 18)
A1 = AB + A – 400g (26)
Remark: Parsing the traverse in the mentioned direction, the orientation towards
the front side will results as sum of the orientation towards the back side and the
horizontal angle between the two sides; if by summing 400g is exceeded, than this should
be subtracted from the sum.
1A = A1 + 200g (27)
Remark: the inverse orientation Ji will results as sum of the direct orientation
iJ and 200g; the same specification for exceeding 400g in the sum. With these
specifications:
12 = 1A + 1 – 400 g
21 = 12 + 200 g
2C = 21 + 2 – 400 g (28)
C2 = 2C + 200 g
cCD = C2 + C – 400 g
cCD is the value of the ending orientation obtained from the calculus.
c) COMPUTING ERRORS, CORRECTIONS
p: reading precision of the theodolite;
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n: number of stations.
c1) ERROR OF CLOSING DISCREPANCY ON THE ORIENTATION
e = cCD - CD (29), condition e < T = pn
Remark: the errors are differences between the erroneous value (affected by
errors) and the correct value (initially given).
c2) TOTAL CORRECTION ON THE ORIENTATION
C = - e
Remark: logical C + e = 0
c3) UNITARY CORRECTION ON THE ORIENTATION
CCu = ------ (30)
n
n: the number of measured horizontal angles, the number of stations.
Remark: the weight factor is the same, because the same device was used, in the
same circumstances, with the same methods (number of measurements), with the same
computational methods for determining the final values of the measured elements, and
the operations were performed by the same devices.
d) COMPENSATING ORIENTATIONS
A1 = A1 + 1 x Cu
12 = 12 + 2 x Cu
2C = 2C + 3 x Cu
CD = cCD + 4 x Cu = CD (COMPULSORY VERIFICATION)
e) COMPUTING COARSE RELATIVE COORDINATES (figure 19)
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It is known that:
XiJ = DiJ cosiJ (31)
YiJ = DiJ siniJ
and will result:
XA1 = DA1 cosA1
YA1 = DA1 sinA1
X12 = D12 cos12
Y12 = D12 sin12
X2C = D2C cos2C
Y2C = D2C sin2C
f) COMPUTING THE ERROR OF CLOSING DISCREPANCY ON THE
COORDINATES
C
eX = XiJ - XAC
A
C
eY = YiJ - YAC
A
C
Where XiJ = XA1 + X12
+ X2C
A
C
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Where YiJ = YA1 + Y12
+ Y2C
A
XAC = XC
– XA
YAC = YC
– YA
Remark: The measuring margins should be respected.
e = e2X + e2
Y, the total closing discrepancy error (32)
C
iJC
A
T = 0.003 iJ + ------------ (33)
A 500
the closing tolerance on coordinates, where
C
iJ = DA1 + D12 + D2C m (34)
A
g) COMPUTING THE CORRECTIONS IN RELATIVE COORDINATES
g1) CAX = - eAX total corrections
CAY = - eAY (35)
g2) Unitary correction:
CAX
CuAX = ------------- C
iJ
A (36)
CAY
CuAY = ------------- C
iJ
Ah) COMPENSATING RELATIVE COORDINATES
119
XA1 = XA1
+ CuAX · DA1
YA1 = YA1
+ CuAY · DA1
X12 = X12 +
CuAX · D12
Y12 = Y12 +
CuAY · D12 (37)
X2C = X2C +
CuAX · D2C
Y2C = Y2C +
CuAY · D2C
CONTROLLING COMPUTATIONS
C
XiJ = XAC
A
C
YiJ = YAC
A
i) COMPUTING THE ABSOLUTE COORDINATES OF THE TRAVERSE
COORDINATES
X1 = XA + XA1
Y1 = YA + YA1
X2 = X1 + X12
Y2 = Y1 + Y12
VERIFICATION:
XCC = X2 + X2C = XC
YCC = Y2 + Y2C = YC (38)
Remark: the computation of the heights of the points is performed in the
following way:
- Coarse relative heights
ZA1 = DA1tgA1
Z12 = D12tg12 (39)
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Z2C = D2Ctg2C
- Closing discrepancy error on heights:
C
eZ = ZiJ - ZAC (40)
A
- Correction on relative heights:
CZ= - eZ (41)
- Unitary correction:
CZ
CuZ = ----------- (42) C
YiJ A
- Compensating relative heights:
ZA1 = ZA1
+ CuZ · DA1
Z12 = Z12
+ CuZ · D12
Z2C = Z2C
+ CuZ · D2C
- Computing absolute heights:
Z1 = ZA
+ ZA1
Z2 = Z1
+ Z12(43)
- Verification:
ZCC = Z2 + Z2C = ZC, where ZC is the height of the point C, from the initial
data.
2. COMPUTING TRAVERSES WITH TWO ENDS AND ONE STARTING
ORIENTATION
The same field and computational steps are parsed, until point c of the previous
case, because we have no ending orientation.
Therefore, steps c, d are not parsed.
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Having the coarse values of the orientations, we move to the computation of step
e and we parse the same computational steps until the end, including the one for the
heights Zi.
3. COMPUTING TRAVERSES WITH TWO ENDS AND ONE ENDING
ORIENTATION
This case is treated as the previous one, computing the coarse orientations from C
towards A: C2 = CD - C, 21 = 2C - 2, 11 = 12 - 1.
4. COMPUTING TRAVERSES WITH TWO ENDS, WITH NO KNOWN
ORIENTATIONS (MINING TRAVERSE)
A and C are existing bearing points, (XA, YA), (XC, YC) are known, 1, 2, 3, …
are new bearing points, (X1, Y1), (X2, Y2), (X3, Y3) are required, (i, i, LiJ) will be
measured, data processing in the field will be performed as in the first presented case.
Because we have no starting orientation, and no ending orientation, we have no
possibility to compute any orientation.
We apply a preliminary computation method:
- We assume that:
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P11 = 100g00c00cc
XPA = 1000.000 m (44)
YPA = 1000.000 m
This allows us to compute:
P1A = P
P1 + 200g
P12 = P
P1 + 1 - 400g
P21 = P
12 + 200g
P23 = P
21 + 2 - 400g
P32 = P
23 + 200g
P3C = P
32 + 3 - 400g
- Preliminary coordinates:
XPA1 = DA1 · cos P
A1
YPA1 = DA1 · sin P
A1
XP12 = D12 · cos P
12
YP12 = D12 · sin P
12
XP23 = D23 · cos P
23
YP23 = D23 · sin P
23
XP3C = D3C · cos P
3C
YP3C = D3C · sin P
3C
- Preliminary absolute coordinates:
XP1 = X
PA+ XP
A1
YP1 = Y
PA+ YP
A1
XP2 = X
P1+ XP
12
YP2 = Y
P1+ YP
12
XP3 = X
P2+ XP
23
YP3 = Y
P2+ YP
23
XPC = X
P3+ XP
3C
YPC = Y
P3+ YP
3C
- The orientation between the assumed bearing points will result from:
123
YPAC YP
C - YPA
tgPAC = ------------ = -------------- (45)
XPAC XP
C - XPA
- The orientation between the bearing points, from the initial data, will be:
YAC YC - YA
tgAC = ------------ = -------------- (46) XAC XC - XA
- The difference on orientations:
= PAC - AC (47)
will be the rotation angle of the entire system arbitrarily chosen, therefore the correct
orientations will be:
A1 = PA1 +
12 = P12 +
23 = P23 +
3C = P3C +
- From this step, assuming that the orientations computed before are the
compensated (correct) ones, the same steps will be parsed, beginning with
step e, as in the first presented case, including for heights (if it is the case).
SURVEY OF PLANIMETRIC DETAILS
Initially, prepare a schema containing the details measured in the station – see
figure 21.
124
METHOD USED: the method of radiation, hence a method with polar coordinates
(12, D3.12) of the position of the characteristic point, with respect to a support base (e.g.
the position of the characteristic point 12 with respect to the support base 32).
Parsing the steps of details survey in the field will respect the following
specifications:
- The maximal distance bearing point – characteristic point is 100 m;
- The number of points measured from one station should be less than 100;
- The measurement of characteristic points will be performed in right handed
direction, starting from the support base, in one position of the telescope
(position I);
- The first and last aim will be towards the bearing point (e.g. from station 3
towards point 2);
- For each characteristic points, measure the following elements:
- The horizontal angle i;
- The declivity angle of the terrain i;
- The slanted direction LiJ (or, directly, the horizontal distance DiJ).
Distances can be measured directly (with the measuring reel) of indirectly
(tacheometrically or electronically).
OFFICE COMPUTATIONS (e.g. point 12, station 3, base aim 32):
- Reducing the distances at the horizon D3.12 = L3.12 cos 12 (48)
- Computing the altitude difference Z3.12 = L3.12 sin 12 (49)
- Computing the planimetric relative coordinates:
X3.12 = D3.12 cos 3.12 (50)
Y3.12 = D3.12 sin 3.12
125
where 3.12 = 32 + 12 (51)
- Computing the absolute coordinates:
X12 = X3 + X3.12
Y12 = Y3 + Y3.12 (52)
Z12 = Z3 + Z3.12
COMPILING THE PLANIMETRY
It represents the operations by which the bearing points and characteristic points
measured in the surveyed area are repeated on a sheet of paper (tracing paper).
The order of the operations for compiling the plan is the following:
- The layout of the paper sheet for representing the plan is chosen based on the
shape and size of the measured area and on the repeating scale;
- The exterior frame (representing the final contour of the plan) is traced at 1-2
cm from the edge of the paper sheet;
- There are traced the border of the plan and the index containing: drafting date,
implied factors (institutions, persons, the beneficiary of the work), repeating
scale, specifications concerning the measured area (locality, county);
- The graticule of the plan is traced (every 5 cm, or every 10 cm), in the
coordinate system in which the work was performed (X0Y);
- Repeat through Cartesian coordinates the points of the control network and
other points for which these coordinates were computed;
- Repeat through polar coordinates (i, DBENCHMARK.i) the characteristic points,
the angle with the use of a protractor with centesimal gradations, the distance:
DBENCHMARK..i
dBENCHMARK.i = ------------------- (53) N
N is the denominator of the plan scale, with a graduated bar.
Attention: the repeating precision will be 0.1g - 0.2g for angles and 0.1 -
0.2 mm for distances;
- The helping points and lines are erased;
- The details are contoured, uniting the characteristic points with each other,
according to the terrain schema;
126
- The plan is finished: inscriptions, names of natural and artificial details; the
writing is done on the west-east direction, eventually along the drawn details
(for natural details);
- Indicate the direction of the geographic north;
- Compile the legend of the plan, the graphical scale.
The operations mentioned before refer to the manual compilation of the plan, but
nowadays most field operations are performed with the complete topographic station,
data processing is performed automatically, based on some specialized programs, the
compilation of the plan is performed with the computer, with the use of horizontal or
vertical plotters.
CHAPTER VII – LEVELING SURVEYS
THE LEVELING
Studies the methods and instruments that are used for determining the heights
(altitudes, level) of leveling bearing points and leveling characteristic points.
Leveling survey obtains data that completes the topographic plans with data
concerning the relief of the surveyed area, offering a better perception over the its actual
aspect.
127
HEIGHTS, LEVEL SURFACES
The level surface is the surface normal in each point to the vertical direction of
the location (of the plumb-bob wire, the direction of the gravity force).
The zero level surface, on the entire planet, is called GEOID.
The GEOID is the surface of the planet obtained extending the seas and oceans
beneath the continents and removing the dry land.
The zero level surface is particular for each state, for Romania, since 1970, being
the Black Sea (until then, it was the Baltic Sea). Since the sea level varies in time, in
order to designate the origin height, for each country, there is built a FUNDAMENTAL
ORIGIN BENCHMARK for heights. For our country, this benchmark is incrusted in the
dam from Constanta, on the Black Sea shore. This point represents the basis for
computing the heights of all (bearing or characteristic) points on the entire national
territory.
For each bearing (or characteristic) point, a level surface can be defined (e.g. for
A or B figure 1).
The ABSOLUTE HEIGHT represents the distance measured on the vertical
between the zero level surface and the level surface that passes through the computed
points (e.g. ZA, ZB).
The RELATIVE HEIGHT (ALTITUDE DIFFERENCE) represents the distance
measured on the vertical between some two level surfaces (e.g. ZAB).
128
The basic computational relation in LEVELING is:
ZB = ZA + ZAB (1)
Where:ZA is a known height, from previous works;
ZAB is the altitude difference determined by a leveling measurement
method;
ZB is the newly computed height.
THE EFFECT OF THE INFLUENCE OF THE EARTH
CURVATURE AND THE ATMOSPHERIC REFRACTION
Let us take two points A and B on the surface of the Earth and build level surfaces
through these points.
Through A we can build a horizontal (plan) surface that we call APPARENT
LEVEL. At distance D (apparently DAB), the effect of the Earth curvature will be C1 =
PP0.
129
Practically, instead of determining the actual altitude difference ZAB, the
apparent altitude difference Z’AB is determined. From AP0:
(R + C1)2 = R2 + D2 (2)
R2 + 2 RC1 + C21 = R2 + D2 (3)
D2
From where: C1 = ----------- (4)
2R + C1
At the denominator it is insignificant in comparison to R, therefore the relation
becomes:
D2
C1 = ----- (5)
2R
R 6379 km (for Romania), hence for D = 1 km, the correction can exceed 70
mm.
Because of atmospheric refraction, the aim from A towards B experiences a
deviation, going on the AP’ trajectory, a second correction C2 will result, with opposite
sign in comparison with the first one:
D2
C2 = ----- K (6)
2R
K is the atmospheric refraction coefficient, K 0.13 (for Romania)
D2
C = C1 – C2 = ----- (1 - K) (7)
130
2R
C is always positive, and for D = 1 km it can exceed 60 mm.
Hence, the corrected value of the altitude difference will be:
ZAB = Z’AB + C (8)
LEVELING TYPES
We have seen that the element measured in leveling is the altitude difference
Ziy, the height being a computational element (Zy = Zi + Ziy).
The altitude differences can be determined by means of many methods, but in
practice the following ones are used:
- GEOMETRIC LEVELING (figure 3);
- TRIGONOMTRIC LEVELING (figure 4);
and less used:
- PHOTOGRAMMETRIC LEVELING, in which the altitude difference is
determined studying the images of the points using the stereographical
principle;
- AUTOMATIC LEVELING: devices assembled on vehicles that parse a route,
constructing automatically the profile of the terrain.
131
LEVELING NETWORKS
As in the case of planimetry, a leveling control network is built on national level,
representing the base of all leveling surveys in the territory.
The leveling geodetic network consists of 4 orders:
- ORDER I:
- Accidental mean square error 0.5 mm/km of traverse;
- Systematic error 0.5 mm;
- Consists of closed polygons, with lengths up to 1500 km, developed along
the main traffic routes of the country.
- ORDER II:
- Total error less than 5 mm L km;
- Developed through polygons with lengths up to 600 km, along traffic
routes;
132
- Must cover uniformly the entire surface of localities, distributed in such a
way that the distance between them would not exceed 2 km, and 3-5 km
outside localities.
- ORDER III:
- Total error less than 10 mmL km;
- Should cover homogeneously the entire surface of localities, the maximal
distance between benchmarks being 200-800m.
Remark: for the first three leveling orders, for computing heights we should take
into account the lack of parallelism of level surfaces (figure 5).
- The level surfaces are not parallel because the distance between two level
surfaces is maximal at the equator and minimal at the poles;
- The leveling for orders II and III is compulsorily executed on back and forth
routes.
ORDER IV is accomplished through middle geometric leveling traverses,
supported at both ends on higher order points, executed only forth.
DESIGNATING AND SIGNALING LEVELING POINTS
133
The leveling points are designated in the terrain respecting the following
conditions:
- To be solidly built (benchmarks, boundary marks);
- To be placed in stable areas (building wall, stable ground safe of land slides,
settlings, vibrations);
- To allow signaling with a measuring staff, in the moment of measurements.
Leveling benchmarks can be:
- Leveling boundary marks, made of concrete (reinforced concrete), with
metallic coupon with hemispheric head at the upper end, protected at the
surface end (filling or lid);
- Benchmarks placed in the nodes of buildings (for which the settling process
has stopped – practically, buildings older than 10 years), metallic, with
circular or hemispheric head, the upper part having specified height;
- Temporary benchmarks: wood or metallic stakes, similar to those used in
planimetry, points of temporary interest;
- Leveling broaste: passing points, signaled by devices on which the measuring
staff could be placed, and which can be temporarily fixed in the ground (the
lower part can be planted in the ground).
GEOMETRIC LEVELING
It creates a reference horizontal surface during measurements, generated by
moving the TOPOGRAPHIC LEVEL around the vertical axis.
The altitude difference between the two points is computed with respect to the
distances from the measured points to this surface.
MIDDLE GEOMETRIC LEVELING
134
If A is a benchmark of known height and B is a point of unknown height (in
general, a surveyed point).
The absolute height of the point A is known: ZA.
The rod readings a, b are measured (level wire) in order to determine the distances
DSA, DSB and in order to verify the readings on the level wire there can be performed
readings on the stadia hairs (upper-lower), too.
135
The distance device-measuring staff will result from the relation:
Dsi = K.H = K (Si – Ji) (9)
Usually on levels K = 100.
Therefore, the height of the measured point B: ZB is required, and will be
obtained from the relations:
ZAB = a – b (10)
ZB = ZA + ZAB
Or,
Zi = ZA + a (the height of the station horizon) (11)
ZB = Zi - b
The second computational method is preferred in the case when the heights of
many points are computed from a station (leveling radiation). It should be noticed that
using middle geometric leveling eliminates the errors determined by the inclination from
the horizontal of the aim towards the two points (given by the device, incorrect horizontal
136
setting of rigid devices, atmospheric refractions), therefore the use of this method is
recommended whenever it is possible.
END GEOMETRIC LEVELING
When middle geometric leveling cannot be applied, there can be applied this
method, which presents the following disadvantages:
- The height of the instrument in the station can be measured with an
approximate error of ± 5 mm (larger than the reading error of the
measurements a, b, which is around 1-2 mm), error that can be eliminated
applying the method presented in figure 9, point b.
- The errors of the inclination of the aiming axis are not eliminated, and they
affect the results of measurements.
The computations are similar to those presented at middle leveling, for the case
a), replacing a by i in the calculus.
MIDDLE GEOMETRIC LEVELING TRAVERSES
Their purpose is to thicken the leveling control network to the level where there
exists a sufficient number of points of known height in the altimetrically surveyed area,
which are needed to measure the heights of all leveling characteristic points.
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The geometric leveling traverse creates a leveling network (which can coincide
with the planimetric one).
Leveling traverses supported on points from the state leveling network form order
V networks and must satisfy the following conditions:
- The length of the span 10-150 m;
- The aiming radius should not get closer than 0.5 m to the surface of the
ground;
- The devices used should be verified and rectified and should have the
magnifying power of the telescope at least 20X;
- The routes of leveling traverses for order V will be compulsorily supported on
points of order I-IV;
- The length of the traverse should not exceed 10 km;
- Designating the points should be stable, solid, not variable as position in time;
- The route of the traverse should not include abrupt slopes (which require short
spans);
- The device will be protected during measurements against the action of
sunlight (a field umbrella will be used);
- The measuring staffs should be vertical (with the plumb-bob wire or spherical
level of the equipment);
138
- Repeated stations will be performed (at least two for each level), in order to
have a verification of measurements and to improve the measuring precision.
CLASSIFICATION OF GEOMETRIC LEVELING TRAVERSES
A. BY THEIR SHAPE:
- Isolated traverses;
- Linked traverses, forming real leveling networks, by the means of some
common points, called NODES.
B. BY THE MEASUREMENT METHOD:
- Traverses with one horizon (only one station for each level);
- Traverses with two horizons (in each station, after measurement, the station is
refreshed – the device is recoded and the measurements are repeated).
C. BY THE WAY OF SUPPORTING AND DISPOSAL IN THE FIELD
(figure ?)
1. Traverses supported at both ends;
2. Traverses in closed circuit;
3. Floating traverses, supported only at one end;
4. Free leveling networks, not supported on points of known heights;
5. Bounded leveling networks, supported on points of known heights.
Remark: in order to present the methods of geometric leveling in a unitary
manner, we shall denote the points of known height with A, B, …, and the new points
with 1, 2, …, P being the node of the networks for the traverses of type 4 and 5.
MIDDLE GEOMETRIC LEVELING TRAVERSE SUPPORTED AT
BOTH ENDS
Let A and B be points of known heights ZA, and ZB.
The new points of the traverse are 1, 2, 3, and 4, for which the new heights Z 1, Z2,
Z3, Z4 will be computed.
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The route of the traverse is parsed continuously from a point of known height A
towards another point of known height B, successively measuring the points A and 1
from station S1, 1 and 2 from S2, and so on (figure 11).
Each station will be done with the first aim backward (to the left on the schema –
e.g. in station S3, first aim the point 2, then the point 3).
It is required that the values on the measuring staff at stadia hairs to be read, in
order to compute the distance stations – aimed points, needed to establish the weight
element in correction distribution. Thus, if the readings in station S1 will be:
- Toward A: SA1, a1, JA
1 (SA1 the reading at the upper hair, a1 the reading at the
level wire, JA1 the reading at the lower hair);
- We shall use the extreme readings:
SA1 + JA
1
- For verifying the median reading a1 = ------------------- 12 mm; 2
- Computing the distance station S1 – point A: DS1A;
DS1A = KHA = K (SA1 – JA
1) (12)
- Similarly:
DS11 = KH1 = K (S11 – J1
1) (13)
and the weight distance of the first level will be:
140
D1 = DS1A + DS11 (14)
COMPUTING THE TRAVERSE
a) COARSE ALTITUDE DIFFERENCES
DZA1 = a1 - b1
DZ12 = a2 - b2
DZ23 = a3 - b3
DZ34 = a4 - b4
DZ4B = a5 - b5
b) ALTITUDE DIFFERENCE ERRORS
B
e2 = ZiJ - ZAB
A
B
Where ZiJ = ZA1 + Z12 + Z23 + Z34 + Z4B (15)A
ZAB = ZB - ZA (16)
c) TOTAL CORRECTION ON ALTITUDE DIFFERENCES
CZ = - e2
d) UNITARY CORRECTION ON ALTITUDE DIFFERENCES
C2
CuZ = ------------ (17) B
Di
A
B
Where Di = D1 + … + D5 (18) A
e) COMPENSATING ALTITUDE DIFFERENCES
ZA1 = ZA1 + CuZ · D1
Z12 = Z12 + CuZ · D2
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Z23 = Z23 + CuZ · D3
Z34 = Z34 + CuZ · D4
Z4B = Z4B + CuZ · D5
f) COMPUTING ABSOLUTE HEIGHTS
Z1 = ZA + ZA1
Z2 = Z1 + Z12
Z3 = Z2 + Z23
Z4 = Z3 + Z34
calc
VERIFICATION: ZB = Z4 + Z4B = ZB (initially given) (19)
Remark: the closing discrepancy error on altitude differences will be verified to
satisfy the condition:
e2 T2 (20)
Where T2 = e2 D km (21)
ekm is the error per km given by the instructions for the performed measurement
class.
COMPUTING THE LEVELING TRAVERSE IN CIRCUIT
- It is performed in the same way, with the specification that if the ending point
coincides with the starting point, at point b) of computation:
A
e2 = iJ (22) A
Since AA = 0
COMPUTING THE FLOATING LEVELING TRAVERSE
- No compensations are done, since there is no closing element;
- Therefore, there will be applied the computational steps a and f.
142
COMPUTING LEVELING NETWORKS
It is a complex method, which goes beyond the framework of this lecture, these
operations being performed by specialists with level A (networks) certification, using
geodetic methods for measuring and processing data.
LEVELING SURVEY OF SURFACES THROUGH GEOMETRIC
LEVELING
The details and characteristic points of the measured area are identified and one of
the presented methods is applied, depending on the conditions in the field.
LEVELING RADIATION
Based on the principle of middle geometric leveling, from a leveling station Si,
determined the heights of characteristic points from within the area of the station, with
respect to the known heights ZA of a leveling benchmark.
143
Thus, the station will be placed in the gravity center of the measured area, at a
distance of at most 50-100 m from the benchmark of known height ZA.
Perform the rod readings a, b1, …, and compute the heights based on the relations:
i = A + a (23)
1 = i – b1
If the topographic level that is used has a horizontal circle, then the measurement
can be completed with planimetric details concerning the measured points: readings at the
stadia hairs – in order to determine the horizontal distances device-aimed point, and at the
horizontal circle – in order to determine the directions station-aimed point.
Remark: in this last case, the utility of measuring distance and angles is not
stressed out, unless the station and the point of known height have known plan
coordinates or if the point of known height has known plan coordinates and we are
stationing in it (the case of the complete topographic station).
THE METHOD OF SQUARES
It is a method applied for leveling survey of some surfaces that are less rough
(agricultural, horticultural, vegetable gardening, rice paddies, areas intended for some
civil, agricultural, industrial construction works: airports, arenas, industrial complexes,
etc.).
Depending on the degree of roughness of the terrain, its size, the degree of detail
coverage, the necessary precision, surface leveling can be executed through small squares
or large squares.
SURFACE LEVELING THROUGH SMALL SQUARES
144
- It is applied in less rough terrains, with declivities < 5%, visibility from the
gravity center of the terrain over the entire surface, surveyed surface of order
of some hectares (at most 4);
- The sides of the squares will have the order of 5-25 m (5, 10, 20 m optimally);
- Choosing the way to divide the surface into squares, the size of the square
side, the number of squares on an axis and on the perpendicular axis is
performed depending on: the precision required for knowing the relief of the
area, the scale of the plan, the degree of roughness of the terrain;
- The working steps are (figure 13):
- Identify the area of study;
- Build a basis AB along one side of the area, which will be pegged out at
equal distances, obtaining the points 1, 2;
- Levels with graduated horizontal circle or a theodolite are used for
pegging out the squares;
- Trace the point C, pegging out the AC axis with the points 3, 7, 11;
- From B, trace the point D, pegging out the points 6, 10, 14;
- From C, aiming D, peg out the points 15, 16;
- From 15, aiming 1, peg out the interior points 12, 8, 4, 1, etc.
145
Pegging out is done with wood stakes, on which, the number of the point can be
written on the upper part.
- Stationing in the gravity center of the measured area, aim, starting from the
benchmark of known height, which can be found in that area (or close by), in
horizon tour, (or scanning the horizontal surface), all the corners pegged with
squares;
- For verification and for avoiding any confusion, I recommend to perform all
the three readings on the measuring staff (up, level wire, down) and to station
in the close vicinity of a square corner (e.g. 8 or 9);
- The procedure can be repeated, from a new station S’1.
Computing the heights of the square corners will be performed similarly to those
from leveling survey:
- Compute the height of the instrument horizon:
Zi = ZRN27 + a (24)
- Compute the height of the radiated points:
Z1 = Zi - b1 (25)
If, for certain reasons:
- The maximal aim station-aimed measuring staff of 150 m is exceeded;
- Obstacles from the area (vegetation, buildings) impede the aims towards
certain points;
146
- There are too many squares visionable from one station (at most 40) – which
means more than 80 points aimed from one station, then there can be used
other methods for the leveling survey of square corners:
- Traverses in closed circuit with radiations;
- Compensated traverses, combined with radiations, on more routes.
The computation of heights in the case of traverses is done similarly as in the case
of leveling of surfaces through large squares.
SURFACE LEVELING THROUGH LARGE SQUARES
It is performed on larger areas (4-100 ha), choosing sides of 50-200 m (50, 100, or
200 m, optimally).
It can be applied only in the case of plain fields or of terrains with constant
declivity on a direction.
Tracing alignments and pegging out square corners can be done as in the previous
case, but it is recommended to use a theodolite-tacheometer for tracing directions, which
can be used also for tracing directions (a precision of 0.1 0.2 m/100 m is enough).
147
Depending on the number of squares station in the center of each square or in
contour squares (e.g. squares 78.12.13, 17.18.23.22, etc. do not have to be stationed,
because the heights of the corners can be computed from the other squares).
Data processing is done in the following way:
Consider the traverse in closed circuit:
RN17 12345.10.15.20.25.30.29.28.27.26.21.16.11.6. RN17, which is compensated
and computed, computing the heights of the points included in the traverse.
The heights of the other points is computed as in the case of traverses supported at
both ends:
E.g. the traverse 6.7.8.9.10, with previously computed end points 6, 10 and new
points 7, 8, 9, etc.
The surface can be parsed through independent traverses, too, including some of
the square corners through various routes. The heights of the other corners can be
computed by leveling radiation.
For example, if, from station S9, the points 20 and 25 were included by leveling
traverse, and their heights were computed using this method, then the heights of the
points measured from S9, not included in the traverse (in this case 18 and 24), can be
computed through leveling radiation, taking into consideration the known height Z20.
THE PRECISION OF GEOMETRIC LEVELING
For geometric leveling of order IV, the tolerance is T = 20 mmD (km), and for
geometric leveling of order V, the tolerance will be T = 30 mmD (km), where D is
the length of the traverse, in km.
A useful application of surface leveling through small or large squares is
embankment cartogram.
148
Practically, after computing the heights of the graticule corners, which cover the
studied area, different studies can be conducted concerning the arrangement of this
surface.
The arranged terrain presents a leveling of the entire surface (on which a certain
objective will be performed), either as a horizontal platform, or as a platform inclined on
one or more directions.
In all these cases, the heights arranged in each square corner can be computed.
For simplification, we assume that the entire platform will be arranged at a designed
height Zp.
We would like to establish which is the nature (digging – cutting or filling up –
embankment) and volume of embankment works, in order to get from the natural terrain
to the terrain arranged at the height Zp.
After computing the height of each square corner, compute the altitude
differences (the execution height):
Zip = Zp – Zi (26)
If Zip > 0 in the area of that point, we shall have an EMBANKMENT volume
(filling up – codified with E in the schema).
If Zip < 0, then there will be a CUTTING volume (digging – codified with C).
149
After computing the values of the execution heights Zip, then we can compute
the embankment volumes (E/C or E + C) for each square.
The case presented in figure 16a is an integral cutting, because in all the four
corners of the square Zip > 0. In this case, the embankment volume (E) will be computed
in the following way:
Sp = l² (27)
l = the side of the square;
Sp = the surface of the square (in horizontal projection).
Z1P + Z2P + Z7P + Z6P
Z1276 = ------------------------------------- (28)4
E1276 = Z1276 · Sp (29)
150
The value will be written in the E box from the center of the square. (The C box
will remain empty, because we have no digging volume in the area).
The case presented in figure 16b is an integral cutting, because Zip < 0 in all the
four corners. After computing Sp and the mean value ZiJkl, the volume C will result in a
similar manner, the value obtained being written in this case in the C box, and the R box
remaining empty this time.
The case presented in figure 16c, and 16d is more complex, because the natural
surface is situated at heights partially greater, partially lower than the designed height Zp.
In this case, there should be found the position of the boundary line MN, which separates
the E volume and the C volume by an axis of height Zp.
From figure 16d it results:
d’ + d” = l (30)
Z22P d’-------- = ------Z17P d”
equation with two unknowns d’, d”, and dIII, dIV, respectively, for the axis 23.18.
After determining the two distances, we shall compute:
(d’ + dIII)lSC
P = -------------- (31) 2
(d” + dIV)lSE
P = -------------- 2
surfaces afferent to the cutting/embankment for the studied corner:
Z22P + 0 + 0 + Z23P
Z22MN23 = ------------------------------- (32) 4
Z17P + 0 + 0 + Z18P
Z17MN18 = ------------------------------- (33) 4
because ZM = ZN = ZP
The values C and E will similarly result from the relations:
C = SCP · Z22MN23; (34)
151
E = SRP · Z17MN18;
And will be written into the corresponding boxes for the studied square
17.18.23.22.
After completing all the E/C boxes, then we can centralize the data, summing up
on the vertical and then on the horizontal, in the end obtaining the total volume of
embankment and cutting and the difference between them.
It is recommended that:
- The E volume and the total volumes to be as small as possible;
- The two final values to compensate each other (E ~ C).
SURFACE LEVELING THROUGH PROFILES
It is applied in the case of investment works performed on large distances (km,
tens of km), having reduced widths (tens of meters): traffic routes (roads, railways),
hydrotechnical works (channels, arrangements), land reclamation works (irrigation
ditches, draining off, damming up), main ducts (oil, methane gas, water supply, sewer).
The technical documentation necessary for the optimal design of such works
includes:
1. THE GENERAL LOCATION PLAN, SCALE 1:N;
2. THE LONGITUDINAL PROFILE, DISTANCE SCALE 1:N, HEIGHT
SCALE 1:M (M can be N/10, N/20);
3. TRANSVERSAL PROFILES, DISTANCE SCALE = HEIGHT SCALE = 1:P
(P can be equal with M);
152
The general location plan represents a larger area, because for optimal design of
the investment, there should be analyzed more possible routes.
Leveling for collecting the data necessary for drafting the profiles will contain the
following steps:
- Materialize the support benchmarks for leveling RNi in the field, which will
connect to the state geodetic network for leveling.
The number of support benchmarks will be established depending on the length of
the route: one at each end (origin A, destination B), and one for at most 2-5 km,
depending on the roughness degree of the terrain;
- Peg out the characteristic points: declivity changes, route changes, thickening
points (if the distances between the first two categories exceed 50 m);
- Maybe, determine the planimetric position of pegs, through a planimetric
traverse;
- Otherwise, determine only the distance between the pegs;
- Designating pegs will be performed with two stakes (one designating the peg,
the other – the control peg, having written the number of the peg within the
route).
The route is parsed by middle geometric leveling traverses supported (through
RNi benchmarks) at both ends.
Also, radiations are performed in the traverse towards the other points of the
longitudinal profile, which are not included in the route, and towards the points of the
transversal profiles.
The purpose of the operations is to determine:
- The heights of all points of the longitudinal profile (Zi);
- The distances between the pegs of the longitudinal profile (DiJ);
- The heights of all points of the transversal profiles (Zt);
- The distances between the points of the transversal profiles (Dtv);
153
It can be seen that the central point of the transversal profiles will be compulsorily
included in the longitudinal profile (if this is one of the pegs).
Having these data, we can begin drafting the longitudinal profile and the
transversal profiles. (See chapter 8: PLANS AND MAPS).
Remark: this topic is largely discussed in the chapter TOPOGRAPHIC WORKS
FOR DESIGNING TRAFFIC ROUTES in our work ENGINEERING TOPOGRAPHY.
TRIGONOMETRIC LEVELING
Consists in determining the altitude difference between two points, based on the
horizontal (or slanted) distance, measured or known (e.g. from coordinates), between the
two points and the declivity angle of the terrain (alignment) or the closing angle of the
theodolitic telescope.
In the first case (figure 18), the signal from point 1 (Z1 the required absolute
height) will be aimed at the height of the instrument in point A (ZA the known height):
- In this case, the declivity angle of the telescope L will be equal to the
declivity angle of the terrain , and the hypotenuse distance (h) of the formed
triangle (aiming axis, DA1, h), will be equal to Z A1,
h = LA1 sin L = LA1 sin (35)
ZA1 = h = LA1 sin (36)
Z1 = ZA + ZA1 (37)
In the case when we cannot aim at the height of the instrument (i) or in the case of
trigonometric leveling on large distances (case in which [DAB = X²AB + Y²AB]), the
signal from B is aimed at a measured height s.
154
If DAB is electronically measurable or can be deduced, then:
h + i = ZAB + s (38)
ZAB = h + (i – s) (39)
h = DAB tg L (40)
ZAB = DAB tg L + (i – s)
ZB = ZA + ZAB (41)
If L AB , i, s, L are measured, the following equations will be taken into account:
ZAB = ZAB tg L + (i – s) (42)
L²AB = D²AB + Z²AB
with two unknowns ZAB, DAB.
155
In the case when the declivity of the terrain is negative ( < 0) and the inclination
of the telescope is negative (L < 0) (figure 20):
ZAB + i = h + s (43)
ZAB = h + (s – i) (44)
h = DAB tgL (45)
ZAB = DAB tgL + (s – i) (46)
And if LAB, i, s, are measured, then apply the system (46).
In this case ZB = ZA - ZAB (47)
In the case when D > 500 m, there appears the influence of the Earth globosity
and atmospheric refraction error, which will be corrected with the value:
D²C = (1 - K) -------- (48)
2R
where K: the atmospheric refraction coefficient (0.13 for Romania);
R: the average radius of the Earth (6379 for Romania).
! c > 0 and it is added to ZiJ.
156
TRIGONOMETRIC LEVELING TRAVERSES
The development of modern methods for precise measuring of distances by
electronic means has extended the applicability scope of some methods, less used before.
Among them is the method presented in the sequel, which has the advantage of
performing planimetric and leveling measurements simultaneously, being a combination
of planimetric traverse and leveling traverse.
There are given: A, B, C, and D mix topographic benchmarks;
(XA, YA, ZA); (XB, YB, ZB); (XC, YC, ZC); (XD, YD, ZD);
ZB and ZD do not have to be necessarily known, because they do not intervene in
the computation.
157
In each station “J” with aims towards the points “i” (backward) and “k” (forward),
measure the following elements:
iJ: the height of the instrument in the station;
Di, sk: the aiming height of range poles (benchmarks, reflectors) from the points
“i” and “k”;
DJi, DJk: the horizontal distances (electronically or LJi, LJk directly);
Ji, Jk: the declivity angles of the telescope of the device towards the two points;
J: the horizontal angle formed by the directions Ji and Jk.
Remark: in the case of complete topographic stations, after horizontal setting and
centering of the device in station J, introduce iJ, si, sk, the names of points i, J, k, all other
data being automatically collected after aiming the two points.
Data processing:
1. PROCESSING MEASURED DATA
a) HORIZONTAL DISTANCES:
DiJ + DJi
DJi = ------------ (50) 2
b) HORIZONTAL ANGLES: the average of the two positions (position I,
position II);
c) VERTICAL ANGLES: by computation (the average of the two positions),
the vertical angle (declivity angle of the telescope) will be used for
computing the altitude difference Z Ji, and Z Jk, respectively.
Thus: ZJi = DJi tgJi + (iJ - si) (51)
And the corresponding value ZiJ:
ZiJ = DiJ tgiJ + (ii – sJ) (52)
The most likely value is:
ZiJ - ZJi
ZiJ = -------------------- (53)2
since ZiJ - ZJi.
158
Having the values (Ji, DJi), we can begin data processing for the planimetric part
(see PLANIMETRIC TRAVERSE SUPPORTED AT BOTH ENDS).
Having the values (ZJi, DJi), the leveling part can be compensated, using the
computational method from MIDDLE GEOMETRIC LEVELING TRAVERSE
SUPPORTED AT BOTH ENDS.
In the end, the coordinates of the measured points will be obtained: (XJ, YJ , ZJ).
TRIGONOMETRIC LEVELING RADIATION
Together with the development of complete topographic stations, this method
received maximal importance, because it is fast, precise, easy.
The method can be applied simultaneously or separately from trigonometric
traverse.
In the case when radiation is performed simultaneously with traversing, first all
the traverse data will be recorded, and then will detail measurement be performed.
Detail measurement is performed in non-compensated horizon tour, starting with
the back base, in position I of the device.
159
The following data is collected for each radiated point:
- The reading on the vertical circle (for computing the declivity angle of the
telescope);
- The reading on the horizontal circle (for computing the horizontal angle 1);
- The horizontal distance DA1 (electronically or directly);
- The aiming height of the signal (if the measurement if electronically
performed, the height will be constant or equal to i);
The previously mentioned data are sufficient for computing:
- Polar coordinates (1, DA1) for repeating the point on the plan, Z1;
- And/or Cartesian coordinates (X1, Y1) and Z1, for automatic repeating.
TACHEOMETRIC LEVELING
Before complete topographic stations were developed, tacheometric survey of
details performed simultaneously for planimetry and leveling was done through its
methods: diagram tachymetry and stadimetric tachymetry with vertical measuring staff,
the most frequent procedure applied for measuring terrestrial surfaces in order to compile
a topographic map or plan.
Mainly, besides the way to obtain the primary elements: horizontal distances and
altitude differences (discussed in detail in the chapter concerning tacheometers as
topographic instruments), this method represents, in fact, a radiation supported on one
base (side or traverse, e.g. AB or AC), measuring the characteristic points from the area
in non-compensated horizon tour.
- With the use of the graphical scale, actual (field) values of some distances
presented in the plan can be determined or distances can be repeated on the
plan scale, on the MAP/plan;
- The method consists in comparing a distance obtained with the distance gauge
on the map/plan to the graphical scale, placing one of its ends on one of the
gradations of the base, and the other end on the talon, the distance resulting as
the number of the two graphically determined values (figure 1/chapter VIII).
160
In the case of TRANSVERSAL GRAPHICAL SCALE (figure 2/chapter VIII), a
differentiated talon being used, the precision obtained is tens of times better than in the
previous case.
GRAPHICAL PRECISION OF TOPOGRAPHIC PLANS
It is recommended that the measuring/repeating precision of a distance from/on a
map or plan to be:
e = ± 0.1 ÷ ± 0.2 mm (54)
e = graphical error.
The graphical precision of the map/plan will be expressed:
Ps = ± e · n · 10–3
n = the scale denominator of the map/plan;
Ps – allows choosing the plan scale depending on the size and shape of the details
that will be represented.
CLASSIFICATION OF MAPS AND PLANS
The scale on which topographic plans are drafted varies within the interval 1:100
÷ 1:10,000, therefore the plans can be:
- Basic topographic plans (1:2000; 1:5000; 1:10,000), which are plans drafted
on the entire territory of the country, in one cartographic projection system;
- Special topographic plans, with different distances – used especially in
investments.
The maps can be:
- Topographic maps, performed on large scale (n < 100,000), from among
which the basic map of the country, on the 1:25,000 scale (with extension in
some area to 1:5000);
- General topographic maps (1:20,000 – 1: 1,000,000);
- Geographic maps (n > 1,000,000).
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TOPOGRAPHIC SYMBOLS
Details representation, in the case of topographic plans, is performed by
geometrizing (replacing with characteristic points), repeating on a horizontal projection
plan and scaling down. The image obtained will be similar with that of the represented
detail.
In the case of topographic maps, their content of natural and artificial details is
graphically expressed by symbols.
Symbols should be illustrative (that is, to suggest the nature of the presented
element), easy to draw, explicit.
For PLANYMETRY, the symbols are:
- Contour symbols, used for representing the contour of the represented detail,
without other details concerning the position or size of details from within the
represented contour (e.g. forests, orchards, waters, etc.);
- Scale symbols, which indicate exactly the position on the map of a detail, in
its axis, without specifying the contour or any information concerning the
content of the detail (e.g. communes, towns, churches, etc.);
- Explicative symbols, which give details concerning the nature of the
represented elements (e.g. the nature of the detail is specified inside the
contour used to represent an orchard: the species and the average size of
trees).
LEVELING SYMBOLS
Are used to represent the relief on the map or plan (in general, contours, nuances,
shades used to suggest the relief, also specifying details concerning them: heights, the
shape in plan and space).
REPRESENTING RELIEF
The main method for representing the relief, a simple, explicit, suggestive
method, is the method of CONTOURS.
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The contour represents the intersection of the terrain with a horizontal section
plan, practically the curve that unites all the points having the same height in the field.
In order to homogenously and coherently represent the relief, the contours are
equidistant, that is, between the horizontal section plans there is an equal distance E,
called the EQUIDISTANCE (of the contour).
The equidistance is equal to an integer multiple of meters: 1, 2, 5, 10, 10, 50, etc.
Choosing the size of E depends on the nature of the terrain (the roughness degree)
and on the plan scale (e.g. mountainous terrain, scale 1:25,000, E = 2 m, plain E = 5 or 10
m).
The equidistance E, scale down on the plan scale is:
e = E · n (55)
e – the graphical equidistance.
The contours can be:
- Normal contours, traced in a continuous and thin line, at the equidistance E,
on the entire plan or map;
- Main contours, traced in bold at 5 E, which will be connected to the state
geodetic network for leveling.
The number of support benchmarks will be established depending on the length of
the route: one at each end (origin A, destination B), and one for at most 2-5 km,
depending on the roughness degree of the terrain;
- Peg out the characteristic points: declivity changes, route changes, thickening
points (if the distances between the first two categories exceed 50 m);
- Maybe, determine the planimetric position of pegs, through a planimetric
traverse;
- Otherwise, determine only the distance between the pegs;
- Designating pegs will be performed with two stakes (one designating the peg,
the other – the control peg, having written the number of the peg within the
route).
The route is parsed by middle geometric leveling traverses supported (through
RNi benchmarks) at both ends.
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Also, radiations are performed in the traverse towards the other points of the
longitudinal profile, which are not included in the route, and towards the points of the
transversal profiles.
The purpose of the operations is to determine:
- The heights of all points of the longitudinal profile (Zi);
- The distances between the pegs of the longitudinal profile (DiJ);
- The heights of all points of the transversal profiles (Zt);
- The distances between the points of the transversal profiles (Dtv);
It can be seen that the central point of the transversal profiles will be compulsorily
included in the longitudinal profile (if this is one of the pegs).
Having these data, we can begin drafting the longitudinal profile and the
transversal profiles. (See chapter 8: PLANS AND MAPS).
Remark: this topic is largely discussed in the chapter TOPOGRAPHIC WORKS
FOR DESIGNING TRAFFIC ROUTES in our work ENGINEERING TOPOGRAPHY.
CHAPTER VIII – PLANS AND MAPS
8.1. THE ELEMENTS OF PLANS AND MAPS
DEFINITIONS
THE TOPOGRAPHIC MAP – standard representation of some large surfaces,
with little details, presenting a general view of that surface of terrain, a generalized image
on reduced scale, taking into account the terrestrial curvature.
THE TOPOGRAPHIC PLAN – standard representation of some small surfaces,
whose details, projected on a horizontal plan, are presented reduced and proportional,
without taking into account the terrestrial curvature, on a large scale.
SCALES
The scale represents the constant ration between a distance d iJ between the points i
and J represented on a map/plan and its correspondent DiJ in the field.
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NUMERICAL SCALES
d = 1 (1) D n
n: the denominator of the plan scale
(e.g. large scale 1:1000 – one mm on the plan corresponds to 1000 mm on the
terrain, that is, to 1 m, small scale 1:100,000 – one mm on the plan corresponds to
100,000 mm on the terrain, that is, to 100 m).
GRAPHICAL SCALES
It is drawn on the MAP/PLAN, being a graphical representation of the numerical
scale.
SIMPLE GRAPHICAL SCALE (figure 1)
- Actual (field) values of some distances presented in the plan can be
determined or distances can be repeated on the plan scale on the map/plan
with the use of the graphical scale;
- The method consists in comparing a distance obtained with the distance gauge
on the map/plan to the graphical scale, placing one of its ends on one of the
gradations of the base, and the other end on the talon, the distance resulting as
the number of the two graphically determined values (figure 1).
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In the case of the TRANSVERSAL GRAPHICAL SCALE (figure 2), a
differentiated etalon is used, therefore the precision obtained is tens of times better than
in the previous case.
THE GRAPHICAL PRECISION OF TOPOGRAPHIC PLANS
It is recommended that the measuring/repeating precision of a distance from/on a
map or plan to be:
e = ± 0.1 ÷ ± 0.2 mm (2)
e = graphical error.
The graphical precision of the map/plan will be expressed:
Ps = ± e · n · 10–3
n = the scale denominator of the map/plan;
Ps – allows choosing the plan scale depending on the size and shape of the details
that will be represented.
CLASSIFICATION OF MAPS AND PLANS
The scale on which topographic plans are drafted varies within the interval 1:100
÷ 1:10,000, therefore the plans can be:
- Basic topographic plans (1:2000; 1:5000; 1:10,000), which are plans drafted
on the entire territory of the country, in one cartographic projection system;
166
- Special topographic plans, with different distances – used especially in
investments.
The maps can be:
- Topographic maps, performed on large scale (n < 100,000), from among
which the basic map of the country, on the 1:25,000 scale (with extension in
some area to 1:5000);
- General topographic maps (1:20,000 – 1: 1,000,000);
- Geographic maps (n > 1,000,000).
TOPOGRAPHIC SYMBOLS
Details representation, in the case of topographic plans, is performed by
geometrizing (replacing with characteristic points), repeating on a horizontal projection
plan and scaling down. The image obtained will be similar with that of the represented
detail.
In the case of topographic maps, their content of natural and artificial details is
graphically expressed by symbols.
Symbols should be illustrative (that is, to suggest the nature of the presented
element), easy to draw, explicit.
For PLANYMETRY, the symbols are:
- Contour symbols, used for representing the contour of the represented detail,
without other details concerning the position or size of details from within the
represented contour (e.g. forests, orchards, waters, etc.);
- Scale symbols, which indicate exactly the position on the map of a detail, in
its axis, without specifying the contour or any information concerning the
content of the detail (e.g. communes, towns, churches, etc.);
- Explicative symbols, which give details concerning the nature of the
represented elements (e.g. the nature of the detail is specified inside the
contour used to represent an orchard: the species and the average size of
trees).
167
LEVELING SYMBOLS
Are used to represent the relief on the map or plan (in general, contours, nuances,
shades used to suggest the relief, also specifying details concerning them: heights, the
shape in plan and space).
REPRESENTING RELIEF
The main method for representing the relief, a simple, explicit, suggestive
method, is the method of CONTOURS.
The contour represents the intersection of the terrain with a horizontal section
plan, practically the curve that unites all the points having the same height in the field.
In order to homogenously and coherently represent the relief, the contours are
equidistant, that is, between the horizontal section plans there is an equal distance E,
called the EQUIDISTANCE (of the contour).
The equidistance is equal to an integer multiple of meters: 1, 2, 5, 10, 10, 50, etc.
168
Choosing the size of E depends on the nature of the terrain (the roughness degree)
and on the plan scale (e.g. mountainous terrain, scale 1:25,000, E = 2 m, plain E = 5 or 10
m).
The equidistance E, scale down on the plan scale is:
e = E · n (3)
e – the graphical equidistance.
The contours can be:
- Normal contours, traced in a continuous and thin line, at the equidistance E,
on the entire plan or map;
- Main contours, traced in bold at 5 E. On them is written the value of the
height that they represent.
- Auxiliary contours, traced in discontinuous lines, at ½ E, in the case when E is
too large to correctly present the represented relief;
- Accidental contours, traced in discontinuous lines, at ¼ E, to represent some
agglomerated, rough relief areas.
Figure 4 presents some relief forms, represented by contours.
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8.2. USING MAPS AND PLANS
1. DETERMINING THE GEOGRAPHIC COORDINATES OF A POINT ON THE
MAP
The left lower corner of the map (figure 5) has the values of geographic
coordinates, latitude, longitude, written on it, from which the representation of the
area is started.
170
In this case: 0 = 4500’00”;
0 = 2425’00”.
Interpolating, we can determine the geographic coordinates of any point on the
map. Hence, for A:
A = 4500’00” + 1’ + ”A, where
dA
” = -------- · 60”, and A = 2425’00” + 1’ + ”A, respectively, where d0
dA
”= -------- · 60” (4)d0
2. DETERMINING THE CARTESIAN COORDINATES OF A POINT ON THE
MAP/PLAN
We proceed in a similar manner, projecting the point on the coordinate axes,
towards the closest graticule left/lower corner (M).
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XA = XM + X
YA = YM + Y
dxA
X = -------- X0 (5)dx
0
dyA
Y = --------- Y0 (6)dy
0
with respect to the map/plan scale 1:n.
The significance of the notations results from figure 6.
In the case of high precision measurements, the distortion in time of the paper of
the plan/map, expressed on both directions (X and Y), should be taken into account.
D D
Kx = -------; Ky = ------- (7) dx
0 dy0
where Dx0 = dx
0 · N; Dy0 = dy
0 · N (8)
and D is the theoretical distance that should be between the lines of the graticule.
In this case:
dxA
X = Kx ------- X0 (9) dx
0
dyA
Y = Ky ------- Y0 (10) dy
0
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3. REPEATING A POINT ON THE MAP/PLAN THROUGH CARTESIAN
COORDINATES
Repeating a point A of coordinates XA, YA on the map or plan is the inverse
operation of determining the Cartesian coordinates. Compute:
XMA = XA - XM (11)
YMA = YA - YM
where M is the left/lower graticule corner that is closest to the point A.
XMA YMA
Then: dxA = ---------; dy
A = ----------; n n
n: the plan scale denominator.
Drawing perpendiculars from the graticule axis towards the values dxA, dy
A, the
point A will result at their intersection.
Attention: all graphical operations of measuring or repeating on the map or plan
will respect the graphical precision.
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4. DETERMINING THE HORIZONTAL DISTANCE BETWEEN TWO POINTS
ON THE MAP/PLAN
a. Graphical method (figure 7)
Measure the distance dAB on a map/plan sheet and compute the equivalent in the
field:
DAB = dAB · n (12)
b. Analytical method (figure 7)
It can be seen that:
DAB = X2AB + Y2
AB (13)
Where XAB = XB – XA, YAB = YB – YA (14)
5. DETERMINING THE ORIENTATION OF A DIRECTION ON THE
MAP/PLAN
a. Graphical method (figure 7)
The orientation can be obtained directly by measuring with the protractor
(sexagesimal or centesimal). The determining error reaches values of 10’ – 20’.
b. Analytical method (figure 7)
From the coordinates of the points:
YAB
tg AB = ---------- (15) XAB
6. THE ORIENTATION IN THE FIELD OF MAPS AND PLANS
It can be performed in two ways:
- Based on the details from the terrain, for example orienting the map with the
use of the represented detail (e.g. railway) along the detail from the terrain;
174
- With the use of the compass, orienting the 0X direction on the map/plan on
the direction of the magnetic north indicated by the compass index.
7. DETERMINING SURFACES ON MAPS/PLANS
NUMERICAL METHODS
GEOMETRIC METHODS
175
These methods are used in the case when the surface can be divided up into
known geometric shapes (see figure 9), usually triangles, and we apply the known
relations for each area:
S = p(p-a)(p-b)(p-c) (16)
a + b + cwhere p = -------------, the semi-perimeter of the triangle,
2
a, b, and c are the sides of the triangle, or
B · IS = -------- (17) 2
B: the base, I: the height of the triangle.
TRIGONOMETRIC METHODS
Are used when sides and angles of the triangle are known, the area resulting from
one of the relations:
bc ca abS = ------ sin A = ------- sin B = -------- sin C 2 2 2
THE ANALYTICAL METHOD
A relation for the analytical computation of surfaces from maps or plans will be
proven, the condition being that the surface should be polygonal (or polygonable) and
that the Cartesian coordinates of all apexes should be known.
176
The relation will be proven on the surface of a triangle and then it will be
generalized. It can be seen that:
S123 = Sy112y2 + Sy223y3 – Sy113y3 (18)
(x2 + x1) (y2 - y1) (x2 + x3) (y3 – y2) (x1+ x3) (y3 – y1)
S123 = ------------------------- + ------------------------ - ------------------------
2 2 2
S123 = 1/2(x2 y2 - x2 y1 + x1 y2 – x1 y1 + x2 y3 - x2 y2 + x3 y3 – x3y2
- x1y3 + x1y1 – x3y3 + x3y1) = 1/2 x1(y2 – y3) + x2 (y3 – y1)
+ x3(y3 – y2)
It can be seen that 3 is after 2 (2+1), 1 is before 2 (2-1), and if we replace 2 with i
then we obtain a general relation:
3S123 = 1/2 Xi (yi+1 – yi -1) (19)
1
which, for a given number n of apexes of closed polygon, whose area is computed,
becomes:
nS = 1/2 Xi (yi+1 – yi -1) (20)
1
or its equivalent:
3S123 = 1/2 Yi (xi-1 – xi +1) (21)
1
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The contour of the polygon will be parsed in right-handed direction, starting from
one arbitrarily chosen apex, denoted with “i”.
Similar relations can be obtained using determinants, knowing that:
x1 y1 1
2 S123 = x2 y2 1
x3 y3 1
GRAPHICAL METHODS
If we known sides/angles of the geometric shapes that compose the surface whose
area has to be computed, then there exists the possibility to graphically measure these
values and then to apply geometric or trigonometric relations.
The graphical methods that use parallels or squares are fast, the precision being in
strong correlation with the distances between the parallels/sides of the squares.
In the case of the method of parallels (figure 11), the surface S is covered on the
map/plan with a network of parallels (on a tracing paper) and the distances li are
measured. If a is the distance between the parallels, the plan/map scale 1:n, then:
A = a · n (22)
Li = li · n
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L1 · A
It can be seen that: S1 = ---------- (if it can be approximated this way)
2
(L1 + L2) · A
S2 = -------------------- (23)
2
Ln · A
Sn = ------------- (if it can be approximated this way)
2
n nFrom where: S = Si = A Li (24)
1 1
In the case when end surfaces cannot be conveniently approximated by triangles
with height A, they are computed separately.
The method of squares (figure 12) is similar, but the surface S is covered by a
network of squares with sides equal to a. Count ni (the number of integer squares
included in the surface), and the parts left outside are grouped by two or three to form
integer squares (as good as possible), obtaining a number of squares np. Therefore, the
total number of squares will be:
N = ni + np (25)
The area of a square will be: Sv = A2 (26)
A = a · n
Thus, the total area will be: S = N · Sv (27)
THE MECHANICAL METHOD
It is used especially to determine the surfaces with sinuous contour, with the use
of a mechanical instrument, called POLAR PLANIMETER (figure 13).
Determining surfaces with the planimeter consists in parsing the perimeter of the
surface in one direction (usually, in right-handed direction), starting from some point on
the contour and ending in it.
When the pole P of the planimeter is in the exterior of the surface that has to be
determined (figure 13a), the surface is obtained from the relation:
179
S = Ks · N = Ks (C2 – C1) (28)
And when the pole of the planimeter is in the interior of the surface (figure 13b),
the surface is obtained by the relation:
S = (C n) Ks
(29)
Where: Ks: the constant of the polar planimeter, which is determined in the
following way:
- Fix the pole P in working position, fix the graver M of the planimeter to a
known radius of the bar and planimeter many times the circle with that radius.
The constant will be:
R2Ks = ----------- (30)
(C2-C1)
R is the radius of the circle whose perimeter was parsed;
C2 , C1 represent the initial and final readings on the bar of the device.
If Ks is obtained as a decimal number, and not as an integer, then the length of the
tracing arm should be adjusted, with a new length L’:
180
K’sL’ = L ------ (31) Ks
Where:Ks, K’s are the constants with and without decimals;
L is the initial length of the tracing arm;
C: is the constant of the planimeter, that is, the surface of the base circle,
depending on the length of the arms;
n = C2-C1
THE PRECISION OF THE METHOD
Ks 0.02 S (cm²)
The tolerance admitted between two planimetry determinations of the same
surface S.
LEVELING PROBLEMS
DETERMINING THE HEIGHT OF A POINT on a map/plan with contours.
Draw the line with the greatest slope, through the point (figure 14) towards the
contours neighbor to the point and measure ’, d.
181
From the figure it results:
Z D’---- = ----- (32) E D
D’ d’ · n d’Or Z = E ------ = E ----------- = E ----- (33)
D d · n d
And the height of the point will be ZP = ZM + Z (34)
where M is the point situated on the contour inferior to the point P.
DETERMINING THE DECLIVITY OF THE TERRAIN BETWEEN TWO
POINTS SITUATED ON A MAP/PLAN
The declivity of the terrain between two points is given by the relation:
ZiJ
p = tg = --------- (35) DiJ
where ZiJ = ZJ = - Zi (36)
DiJ = diJ · n
Percent values are also used:
100ZiJ
p% = 100 tg = -------------- (37) (e.g.: roads, ducts)DiJ
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1000ZiJ
p% = 1000 tg = -------------- (38) (e.g.: railways, subway)DiJ
It can be seen that there exists a relation of converse proportionality between p
and DiJ, which means that the greater the altitude difference between the ends of a small
distance on the map, the more abrupt the terrain is in that area.
Remark: in order to study the declivity along a given alignment, we have first to
section the route into areas with approximately constant declivity with the same sign
(positive or negative) (figure 16).
Therefore, parsing the route from A towards B, we could find four area of
approximately constant declivity:
AC: small positive declivity; (ZC, ZA, large distances between two neighbor
contours);
CF: great positive declivity; (ZF, ZC, small distances);
FD: great negative declivity; (ZD, ZF, small distances);
DB: small negative declivity; (ZD, ZF, large distances).
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TRACING A LINE WITH CONSTANT DECLIVITY BETWEEN TWO
POINTS ON THE MAP OR PLAN
From the declivity relation:
100Ep0% = ---------- (39)
d0 · n
100 Ed0 = -----------
p0% · n
d0: the distance between two neighbor contours, such that the declivity of the line
that unites the two contours, of length d0, to be the required declivity p0%.
The tracing is performed with a compass with the span of the arms equal to d0,
from A to B.
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THE TOPOGRAPHIC PROFILE OF THE TERRAIN BETWEEN TWO
POINTS ON THE MAP/PLAN
After tracing the alignment, by uniting its ends (e.g. A and B), number each
intersection with a contour (1, 2, …), measure the horizontal distances diJ between
neighbor points (dA1, d12, …) and record the height of each point (ZA = 220, Z1 = 221, …)
Having these values, build the profile, on scale:
- For the distance 1:m, usually m = n, where 1:n is the map/plan scale;
- For heights 1:c, usually c = 10m.
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