surveying spcs

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Surveying on NAD83 State Plane Coordinate System

ByDr. Joshua Greenfeld

Surveying Program CoordinatorNJ Institute of Technology

Objective

Modern surveying operations which involves working with GIS/LIS and GPS requires the utilization of a nationwide or at least a statewide reference coordinate system. The most common reference system in the US is The NAD83 State Plane Coordinate System (SPCS).

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Objective

The purpose of the seminar is to provide the Professional Land Surveyor with the necessary information for working with the NAD83 SPCS. It will include a discussion on projecting the earth onto a plane, definitions of NAD83 and computational aspects of SPCS. Practical examples such as traverse computations will be practiced by the seminar participants.

OUTLINE

• Background on State Plane Coordinate Systems- The need- The law

• Definition of Terms- Geodetic Reference System- Geodetic Coordinate Systems- Datum

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OUTLINE• Introduction to map projections

– From the curved earth to a flat map– Classes of map projections– Map projections for surveying

• NAD83– Why replace NAD27?– Characteristics of the new datum– NAD83/9x

OUTLINE• State plane coordinate system

– General– New Jersey’s system– Pennsylvania’s system

• Coordinate computations and conversions– Geodetic to Plane– Plane to Geodetic

• Meridian convergence and Scale Factor

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OUTLINE

• Traverse computation on SPCS– (t-T) correction– Scale factor correction– Final coordinate values for traverse points

• GPS applications

The National Spatial Reference System

NSRS serves as the basis for:• mapping• GIS• navigation• military activities

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The nature of NSRS changed over the years

Name Objective1. New England Datum Local2. NAD 27 Adjustment of North America3. NAD 83 Development of EDM 4. NAD 2000 (ITRF) GPS

Some of the benefits for surveyors to work with SPCS

All surveys correlate to a single reference framework. This means that all surveys, old and new, can be combined seamlessly to a consistent and contiguous mapping project. Points from old and new surveys can be used without the need to recompute the old measurements. Surveyors having numerous projects in a certain area could, theoretically, “cut and paste” different projects to produce a map without compromising the accuracy of the new product.

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Large projects can be surveyed in parallel as independent sections. Although during the time of the execution of the project the different sections are not yet connected physically, they are connected computationally because they all share a common reference framework. As the work progresses, all sections will be connected and the accuracy of the entire project will be maintained throughout.


Data sharing among surveyors is simplified if everyone is working on the same reference system. This means that surveys can be re-packaged and sold for additional profit. Data is a precious commodity in the GIS world. Surveyors have an abundance of spatial information; if it is in a useable form (such as SPCS) it has a market value.

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No point can be considered lost because it can be recovered by its coordinates. If you have the State Plane Coordinates of a point, you can use, for example, GPS to recover it. There is no need to recover points using ties that may have been also been destroyed. (Unless there is a legal issue involved.)


Using SPCS, the earth can be viewed mathematically as a plane. This means that plane geometry and trigonometry mathematics can be used in our computations. One needs only to apply a small, well defined, correction to compensate for the plane approximation. This manual explains what corrections have to be made and provides an example of how to apply them.

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Working with SPCS provides an extra external computation check for our surveys. Loop closures such as a closed traverse check only the inner consistency of the survey. If, for example, there is a systematic scale error in the traverse, it will not be detected by summing up the latitudes and the departures. Only if we tie the traverse to two or more points with given State Plane Coordinate values can this error be realized and corrected. A similar argument holds for the orientation of the traverse. To maintain proper orientation of a traverse we need to tie it to at least two control points with State Plane Coordinates.

NJ SPCS Law

Assembly, No. 21-L.1989, c. 218

This bill changes the system of plane coordinates which comprises the official survey base for the State. The plane coordinates system currently in use is a transverse Mercatorprojection of Clarke's spheroid of 1866, with a central meridian located 74° 40' west from Greenwich. Under the bill, the system will be a transverse Mercator projection of the Geodetic Reference System of 1980, with a central meridian located 74° 30' west from Greenwich. The bill also provides for the use of' the North American Datum of 1983 or the most recently published adjustment by the National Geodetic Survey.

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NJ SPCS Law (cont’)All coordinates of the system shall be expressed in meters rather

than in feet. The bill revises the scale of the coordinate system from 1:40,000 (expressed in feet) to 1:10,000 (expressed in meters). It also provides that standard conversions from metersto feet shall be the adopted standards of the National Oceanic and Atmospheric Administration.

COMMITTEE AMENDMENTSThe committee amended the bill to correct the reference to the

Geodetic Reference System of 1980 and to provide that the New Jersey coordinate system defined by the North American Datum of 1927 may be used concurrently with or in lieu of the system defined by the North American Datum of 1983 for a period of 36 months after the effective date of this act.


In many states such as New Jersey there is a SPCS law.

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GEODESY

Geodesy is one of the oldest sciences

Definition:

The science of determining the size and shape of the earth including its gravity field, in four-dimensional space-time

Different Surfaces In Geodesy

• The topography - the physical surface of the earth• The geoid - the level (equipotential) surface at

mean sea level. In simple terms a level surface is a surface on which water will stand still. Thegeoid is a level surface at elevation zero.

• The ellipsoid - the mathematical surface which approximates the shape and size of the earth, and is used as a reference frame for position computations.

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Surfaces in Geodesy

The Figure of the Earth

Major Axis

Min

or A

xis

ab

abaf −=

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Common Ellipsoids

Name Use a - (meters) 1/f

Clarke 1866 NAD27 6378206.4 294.97869820

GRS 1967 (Old GRS) 6378160.0 298.247167427

GRS 1980 NAD83 6378137.0 298.257222101

WGS-72 Previous GPS 6378135.0 298.26

WGS-84 Current GPS 6378137.0 298.257223563

Geodetic Coordinate System

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Geodetic Coordinate System

a

b

Normal to

ellipso

id

Pole

Equator

GreenwichGreenwich

HH

λ

ϕ

Map projection

• A Map projection is a systematic representation of a round body such as the earth on a flat (plane) surface. Each map projection has specific properties that make it useful for specific purpose.

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The Problem of map projection

• The earth is an irregular surface that has to be represented in terms of a simple but consistent graph.

• Computation of Distances and Directions (Az) are very complicated on spheroid but very easy using X,Y coordinates.

Solution

• 1. Approximate the earth by a sphere (or an ellipsoid)

• 2. Reduce the size of the sphere (globe)• 3. Project the globe into another surface

(developable surface) that can be flattened out

• 4. develop a method for calculating the resultant distortions.

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Distance distortion

Shape distortion

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Distortions

Mercator Cylindrical Equal Area

An ideal map projection must satisfy the following:

1. All distances and areas on the map should have correct relative magnitude.

2. All Azimuths and angles should be correctly shown on the map

3. All Great circles on the earth should appear as straight lines.

4. Geographic latitudes and longitudes of all points should be shown correctly on the map

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An ideal map projection?

It is impossible to satisfy all of these conditions.

So, we devise different map projections for specific purposes. The best map projection is the one that best serves a particular application.

The most common classification of map projections:

Azimuthal projection: Shows correct directions and AZ from one central pointEquidistance projection: shows correct distances from one central point.Conformal projection: Shows correct angles from all points, if a small portion of the earth is projected.Equal area projection: Shows correct areas. Shape is usually very distorted.

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Sphere

MapProjection

Choice of Datum

Scale Reduction

CylindricalProjection

ConicProjectionAzimuthal

Projection

Systematic

No actual projectionconstruction of

(mathematical) graduate

Globe

Ellipsoid

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Classification of map projectionsExtrinsic Problem (Projection Surface)

ObliqueTransverseNormalPosition

Poly superficial

SecantTangentCoincidence

CylindricalConicalPlaneNature

Class Varieties

Classification of map projectionsIntrinsic Problem (Projection Characteristic)

Class Varieties

NavigationThematicTopographicUse

RegionContinentWorldMapping extent

conventionalSemi-Geometric

GeometricGeneration

ConformalEquivalentEquidistanceProperties

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GlossaryMap projection - An orderly system of lines on a plane

representing a corresponding system of imaginary lines on an adopted datum surface, and its mathematical concept.

Developable surface- A developable surface is a simple geometric form capable of being flattened without stretching

Datum - Any numerical or geometrical quantity or set of such quantities which may serve as reference or base for other quantities.

Glossary

Latitude - The angle which the normal to ellipsoid at a point makes with the plane of the equator.Longitude - The angle between the plane of the meridian and the plane of an initial meridian, arbitrarily chosen (Greenwich)Great Circle - A great circle is formed on the surface of a sphere by a plane that passes through the center of the sphere. The arc of a great circle is the shortest distance between points on the surface of the earth.

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Glossary

Meridian - An arc formed by the intersection of the plane containing a point on the earth and its two poles.Central meridian - A meridian that is located in the center of the projected area (used only in some map projection)Parallel - An arc formed by connecting all the points that have the same latitudeStandard Parallel - A specific parallel at which the developable surface intersect the surface of the earth

Glossary

Rhumb Line (Loxodromes)- A rhumb line is a line on the surface on the earth cutting all meridians at the same angle. (not necessarily the shortest distance!)

Linear Scale - Linear scale is the relationship between a distance on a map and the corresponding distance on the earth.

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Glossary

Mapping Equations - Mapping equations are the mathematical relationship between the coordinates of the earth and the coordinates of the map

x = f1(ϕ,λ)y = f2(ϕ,λ)

The simplest mapping equations are:x = λy = ϕ

Glossary

For T.M.x = koN [A+ (1-T+C)A3/6 + (5-18T+T2+72C-58e’2)A5/120]

Where:e’2 = (a2 - b2) / b2

N = a /(1-e2 sin2ϕ)1/2

T = tan2 ϕC = e’2 cos2 ϕA = cosϕ (λ-λo)

y = ...

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Some Map Projections

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Orthographic

Plane ofprojection

Equator

PolarMapmaker selectsNorth or South Pole

EquatorialMapmaker selectscentral meridian

ObliqueMapmaker selectsany point of tangencyexcept alongEquator or at Poles

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Orthographic

• Used for perspective views of the Earth, Moon, and other planets. The Earth appears as it would on a photograph from deep space. Used by USGS in the National Atlas of the United States of America.

• The Orthographic projection was known to Egyptians and Greeks 2,000 years ago.

Orthographic• Directions are true only from center point of

projection. • Scale decreases along all lines radiating from

center point of projection. Any straight line through center point is a great circle.

• Areas and shapes are distorted by perspective; distortion increases away from center point.

• Map is perspective but not conformal or equal area. In the polar aspect, distances are true along the Equator and all other parallels.

• Azimuthal-Geometrically projected onto a plane. Point of projection is at infinity.

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Stereographic

Plane ofprojection

Equator

Point of projection




Stereographic

• Used by the USGS for maps of Antarctica and American Geographical Society for Arctic and Antarctic maps. May be used to map large continent-sized areas of similar extent in all directions Used in geophysics to solve spherical geometry problems. Polar aspects used for topographic maps and charts for navigating in latitudes above 80º.

• Dates from 2nd century B.C. Ascribed toHipparchus.

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Stereographic

• Directions true only from center point of projection

• Scale increases away from center point. Any straight line through center point is a great circle.

• Distortion of areas and large shapes increases away from center point.

• Map is conformal and perspective but not equal area or equidistant.

• Azimuthal-Geometrically projected on a plane. Point of projection is at surface of globe opposite the point of tangency.

Gnomonic

Plane ofprojection

Equator




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Gnomonic

• Used along with the Mercator by some navigators to find the shortest path between two points. Used in seismic work because seismic waves tend to travel along great circles.

• Considered to be the oldest projection. Ascribed toThales, the father of abstract geometry, who lived in the 6th century B.C.

Gnomonic

• Any straight line drawn on the map is on a great circle, but directions are true only from center point of projection.

• Scale increases very rapidly away from center point. • Distortion of shapes and areas increases away from

center point.• Map is perspective (from the center of the Earth onto

a tangent plane) but not conformal, equal area, or equidistant.

• Azimuthal-Geometrically projected on a plane. Point of projection is the center of a globe.

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Mercator

Great distortionsIn high latitudes

Central Meridian(selected by mapmaker)

Rhumbs lines(direction true between any two points

Equator touches cylinder if cylinderis tangent

Reasonably true shapes and distances within 15o of equator

Mercator

• Used for navigation or maps of equatorial regions. Any straight line on map is a rhumb line (line of constant direction).

• Presented by Mercator in 1569.• Directions along a rhumb line are true between

any two points on map, but a rhumb line usually is not the shortest distance between points. (Sometimes used with Gnomonic map on which any straight line is on a great circle and shows shortest path between two points.)

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Mercator

• Distances are true only along Equator, but are reasonably correct within 15º of Equator; special scales can be used to measure distances along other parallels. Two particular parallels can be made correct in scale instead of the Equator.

• Areas and shapes of large areas are distorted. • Distortion increases away from Equator and is

extreme in polar regions. Map, however, is conformal in that angles and shapes within any small area (such as that shown by a USGS topographic map) are essentially true.

Mercator

• The map is not perspective, equal area, or equidistant.

• Equator and other parallels are straight lines (spacing increases toward poles) and meet meridians (equally spaced straight lines) at right angles. Poles are not shown.

• Cylindrical-Mathematically projected on a cylinder tangent to the Equator. (Cylinder may also be secant.)

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Transverse MercatorCentral Meridian selected by mapmakertouches cylinder if cylinder is tangent

Equator

Can show whole Earth, but directions, distances and areas are reasonablyaccurate only within 15o of the central meridian

No straight rhumb lines

Transverse Mercator

• Used by USGS for many quadrangle maps at scales from 1:24,000 to 1:250,000; such maps can be joined at their edges only if they are in the same zone with one central meridian. Also used for mapping large areas that are mainly north-south in extent.

• Presented by Lambert in 1772.• Distances are true only along the central meridian

selected by the mapmaker or else along two lines parallel to it, but all distances, directions, shapes, and areas are reasonably accurate within 15º of the central meridian.

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Transverse Mercator• Distortion of distances, directions, and size of areas

increases rapidly outside the 15º band. Because the map is conformal, however, shapes and angles within any small area (such as that shown by a USGS topographic map) are essentially true.

• Graticule spacing increases away from central meridian. Equator is straight. Other parallels are complex curves concave toward nearest pole.

• Central meridian and each meridian 90' from it are straight. Other meridians are complex curves concave toward central meridian.

• Cylindrical-Mathematically projected on cylinder tangent to a meridian. (Cylinder may also be secant.)

Oblique Mercator

Equator

In this projection, shortest distance between points along line of tangency are straight lines.

No straight rhumb lines

Line of tangencythe great circle that touches cylinder if cylinder is tangent

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Oblique Mercator

• Used to show regions along a great circle other than the Equator or a meridian, that is, having their general extent oblique to the Equator. This kind of map can be made to show as a straight line the shortest distance between any two preselectedpoints along the selected great circle.

• Developed 1900-50 by Rosenmund, Laborde,Hotine et al.

Oblique Mercator

• Distances are true only along the great circle (the line of tangency for this projection), or along two lines parallel to it. Distances, directions, areas, and shapes are fairly accurate within 15º of the great circle.

• Distortion of areas, distances, and shapes increases away from the great circle. It is excessive toward the edges of a world map except near the path of the great circle.

• The map is conformal, but not perspective, equal area, or equidistant. Rhumb lines are curved.

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Oblique Mercator

• Graticule spacing increases away from the great circle but conformality is retained. Both poles can be shown. Equator and other parallels are complex curves concave toward nearest pole. Two meridians 180º apart are straight lines; all others are complex curves concave toward the great circle.

• Cylindrical-Mathematically projected on a cylinder tangent, (or secant) along any great circle but the Equator or a meridian.

• Directions, distances, and areas reasonably accurate only within 15º of the line of tangency.

Robinson

Equator

Central Meridian

Pseudocylindrical or orthophanic ("right appearing") projection

Straight equator, parallels, central meridiancentral meridian is 0.53 as long as equator

Concavemeridianare equally spaced

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Robinson

• Uses tabular coordinates rather than mathematical formulas to make the world "look right." Better balance of size and shape of high-latitude lands than in Mercator, Van der Grinten, or Mollweide. Soviet Union, Canada, and Greenland truer to size, but Greenland compressed.

• Presented by Arthur H. Robin- son in 1963.• Used in Goode's Atlas, adopted for National

Geographic's world maps in 1988, appears in growing number of other publications, may replace Mercator in many classrooms.

Robinson • Directions true along all parallels and along central

meridian. • Distances constant along Equator and other parallels,

but scales vary. • Scale true along 38º N & S, constant along any given

parallel, same alone N & S parallels same distance from Equator.

• Distortion: All points have some. Very low along Equator and within 45º of center. Greatest near the poles.

• Not conformal, equal area, equidistant, or perspective. • Pseudocylindrical or orthophanic ("right appearing")

projection

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Albers Equal Area Conic

Two standard parallels(selected by mapmaker)

Equal area. Deformation of shapes increases away from standard parallels


• Used by USGS for maps show-ing the conterminous United States (48 states) or large areas of the United States. Well suited for large countries or other areas that are mainly east-west in extent and that require equal area representation. Used for many thematic maps. Maps showing adjacent areas can be joined at their edges only if they have the same standard parallels (parallels of no distortion) and the same scale.

• Presented by H. C. Albers in 1805.

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Albers Equal Area Conic • All areas on the map are proportional to the same areas

on the Earth. • Directions are reasonably accurate in limited regions. • Distances are true on both standard parallels.

Maximum scale error is 1.25% on map of conterminous States with standard parallels of 29.5 N and 45.5'N.

• Scale true only along standard parallels. USGS maps of the conterminous 48 States, if based on this projection, have standard parallels 29.5'N and 45.5'N. Such maps of Alaska use standard parallels 55'N and 65'N, and maps of Hawaii use standard parallels 8'N and 18'N.


• Map is not conformal, perspective, or equidistant.• Conic-Mathematically projected on a cone

conceptually secant at two standard parallels.

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Lambert Conformal Conic

Two standard parallels(selected by mapmaker)

Large-scale map sheets can bejoined at edges if they have same standard parallels and scales


• Used by USGS for many 7.5 and 15-minute topographic maps and for the State Base Map series. Also used to show a country or region that is mainly east-west in extent.

• Presented by Lambert in 1772. • One of the most widely used map projections in

the United States today. Looks like the Albers Equal Area Conic, but graticule spacings differ.

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• Retains conformality. Distances true only along standard parallels; reasonably accurate elsewhere in limited regions.

• Directions reasonably accurate. • Distortion of shapes and areas minimal at, but

increases away from standard parallels. • Shapes on large-scale maps of small areas

essentially true.


• Map is conformal but not perspective, equal area, or equidistant. For USGS Base Map series for the 48 conterminous States, standard parallels are 33'N and 45'N (maximum scale error formap of 48 States is 21/2%). For USGS Topographic Map series (7.5- and 15-minute), standard parallels vary. For aeronautical charts of Alaska, they are 55'N and 65'N; for the National Atlas of Canada, they are 49'N and 77'N.

• Conic-Mathematically projected on a cone conceptually secant at two standard parallels.

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Geographic

Gnomonic

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Stereographic

Mercator

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Transverse Mercator (TM)

Robinson

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Albert Equal Area Conic


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The Universal Transverse Mercator (UTM) Projection

The Universal Transverse Mercator (UTM) Projection

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Characteristics of NAD'83

Redefinition - Define a new datum that:• Has a worldwide fit (not localized to best fit North

America)• The center of the ellipsoid is defined at the mass

center of the earth. (moved from Meades Ranch KS to be consistent with satellite systems)

• The size (a) and the shape (1/f) are defined by GRS'80.a = 6,378,137. 0 meters 1/f = 298.257222


Readjustment - Compute a new Least Square solution for old and new observations.

• Simultaneous adjustment of 1,785,772 observations for 266,436 stations.

• Combine classical terrestrial and satellite (space) data

• Mathematical modelling to remove unpredictable distortions in the network.

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Plane Coordinates - For various states (or zones) are derived from a conformal mapping projections that:

• Maximum scale distortion is less than 1:10,000• Cover an entire state with as few zones of a projection

as possible

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Characteristics of NAD'83• Define boundaries of projection zones as an

aggregation of counties– TM - Transverse Mercator (N-S zones) or– LCC - Lambert Conformal Conic (E-W zones)– OTM - Oblique Transverse Mercator (Alaska)

Coordinate values are different from NAD'27 due to:

• Change in datum• Changes as a result of a new adjustment• New mapping (projection) equations (derived

mathematically not empirically) to support 1mm accuracy

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Coordinate values are different from NAD'27 due to:

• Changes in numerical grid value of the origin of each zone

• Changes in mapping constants in some zones (new standard parallel or meridian)

• Azimuth orientation is due North• The use of Metric units rather then Feet

Plane coordinates

Plane Ellipsoid

Slanted (not uniformly) towards the North Pole. All lines pointing to North, converge at the North Pole.

Straight up or parallel to the direction of the North Arrow

North-South Direction

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Plane Ellipsoid

Curved lines (curve C-S’-D)

Straight lines (line C-D)

Distances

C D

CC DDSS

Plane coordinates

Plane coordinates

Plane Ellipsoid

Uneven distances, i.e. the length of an arc of 2° of longitude near the pole is much shorter than 2° of arc near the equator.

Even (same length) distances

Even coordinate differences correspond to:

360° + spherical excess

360°Sum of angles in a quadrilateral

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Most Common Projection for SPCS

Projecting part of an ellipsoid onto a plane

Ellipsoid

Projection Plane

Scale >1

Scale <1

Scale=1Scale=1

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LCC Projection

R sin θ

R c

osθ

P

R RbStandardParallels 158 M

iles or less

158 Miles or less

X

Y

θ

XP

YP

Eo

Scale =1Sc

ale <

1Scale =1

Scale >1

Scale >1

TM Grid

γ

Cen

tral

Mer

idia

n

Scal

e >1

Scal

e <1

Scal

e =1

Scal

e =1

Scal

e >1

Stan

dard

Lin

es

Stan

dard

Lin

es

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Computations with SPCS

1. Direct Problem (ϕ , λ) à(N , E )

2. Inverse Problem (N , E ) à(ϕ , λ)

3. Meridian convergence4. Scale factor

• Manual computation• Software

NAD83 SPCS Zone Constants for New Jersey

ZoneCode 2900

Projection TMCentral Meridian 74° 30’

Scale Factor 1:10,000Grid Origin: ϕ 38° 50’

λ 74° 30’E 150,000N 0

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NAD83 SPCS Zone Constants for Pennsylvania

Zone N SCode 3701 3702

Projection LCC LCCStandard Parallels 40° 53’ 39° 56’

41° 57’ 40° 58’

Grid Origin: ϕ 40° 10’ 39° 20’λ 77° 45’ 77° 45’E 600,000 600,000N 0 0

INTERPOLATION OF LCC NAD 83(Pennsylvania North)

Input Zone Constantsϕ = 41° 25' 23.85" ϕb = 40° 10‘λ = 78° 15' 12.12" λcm = 77° 45'

Rb = 7379348.3668Eo = 600000.No = 0.sin ϕo = 0.66153973381

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Interpolation.R41° 26' 7238680.103R41° 25' 7240531.059

∆R= -1850.956

Computation.γ = ( λcm - λ ) sinϕo = -0° -19' -58.789382"E = R sinγ + Eo = 557923.307 N = Rb - R cosγ + No = 139675.336

SPC from LCC NAD 83 (Tables)(Pennsylvania North)

R RR

= + ⋅ =°41 25 602385 7239795 304

' ". " .

∆

SPC from LCC NAD 83 (Tables)(Pennsylvania North)

Input: E = 557923.307N =139675.336

Computation.tan γ = = -0.00581196

R = = 7239795.304

λ = λcm - = 78° 15' 12.12"

To compute ϕ perform a backward interpolation using R.

R N Nb − −( )cos

0

γ

E ER N Nb

−− −

0

0( )

γϕsin 0

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Direct Problem (ϕ , λ) à(N , E )L = λd - λ0

N = S' + A2 × L2 + A4 × L4 - *2nd order correctionE = E0 + A1 × L + A3 × L3 - *2nd order correctionWhere:

λd - Geodetic Longitude in D°.ddddd (Decimal Degrees)λ0- 74°.5 (NJ’s Geodetic Longitude of C. Meridian)E0- 150,000m (NJ’s Grid Easting Origin Shift)S',A2,A4,A1,A3,-Tabulated coefficients as a function of ϕϕ- Geodetic Latitude in D° M' S“N,E- Grid Northing and Easting (m)

*2nd order corrections for accuracies better than ±0.001m.

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Inverse Problem (N , E ) à(ϕ , λ)

ϕ = ϕ' + B2 × Q2 + B4 × Q4 - *2nd order correction

λ = λ0 + B1 × Q + B3 × Q3 + B5 × Q5 - *2nd order correction

Where:λ- Geodetic Longitude in D°.ddddd (Decimal Degrees)

ϕ- Geodetic Latitude in D°.ddddd (Decimal Degrees)λ0- 74°.5 (NJ’s Geodetic Longitude of C. Meridian)ϕ'- Footpoint Latitude. Tabulated as a function of N.E0- 150,000m (NJ’s Grid Easting Origin Shift)E- Grid Easting (m)B2,B4,B1,B3,B5-Tabulated coefficients as a function of N (Northing)

*2nd order corrections accuracies better than ±0.0001".

QE 150,000.1,000,000.

=−

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Data SheetNational Geodetic Survey,

*********************************************************************** KV2920 DESIGNATION - E 78 KV2920 PID - KV2920 KV2920 STATE/COUNTY- NJ/SOMERSET KV2920 USGS QUAD - RARITAN (1981) KV2920 KV2920 *CURRENT SURVEY CONTROL KV2920

___________________________________________________________________ KV2920* NAD 83(1996)- 40 35 16.39834(N) 074 39 05.85416(W) ADJUSTED KV2920* NAVD 88 - 48.230 (meters) 158.23 (feet) ADJUSTED

KV2920 ___________________________________________________________________

KV2920 X - 1,283,866.440 (meters) COMP KV2920 Y - -4,677,505.732 (meters) COMP KV2920 Z - 4,127,785.929 (meters) COMP KV2920 LAPLACE CORR- 3.83 (seconds) DEFLEC99 KV2920 ELLIP HEIGHT- 14.85 (meters) GPS OBS KV2920 GEOID HEIGHT- -33.40 (meters) GEOID99 KV2920 DYNAMIC HT - 48.208 (meters) 158.16 (feet)

COMP

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Data SheetKV2920 HORZ ORDER - SECOND KV2920 VERT ORDER - FIRST CLASS II KV2920 ELLP ORDER - FOURTH CLASS I KV2920

KV2920. The horizontal coordinates were established by GPS observations KV2920. and adjusted by the National Geodetic Survey in May 1999. KV2920 KV2920. The orthometric height was determined by differential leveling KV2920. and adjusted by the National Geodetic Survey in June 1991. KV2920

KV2920. The X, Y, and Z were computed from the position and the ellipsoidal ht. KV2920

KV2920. The Laplace correction was computed from DEFLEC99 derived deflections. KV2920

KV2920. The ellipsoidal height was determined by GPS observations KV2920. and is referenced to NAD 83. KV2920

KV2920. The geoid height was determined by GEOID99. KV2920

KV2920. The dynamic height is computed by dividing the NAVD 88 KV2920. geopotential number by the normal gravity value computed on the KV2920. Geodetic Reference System of 1980 (GRS 80) ellipsoid at 45 KV2920. degrees latitude (g = 980.6199 gals.). KV2920

KV2920. The modeled gravity was interpolated from observed gravity values.

Data SheetKV2920; North East Units Scale Converg. KV2920;SPC NJ - 194,798.230 137,165.060 MT 0.99990203 -0 05 55.1 KV2920;UTM 18 - 4,493,068.719 529,480.530 MT 0.99961070 +0 13

36.0 KV2920KV2920 STATION DESCRIPTION KV2920 KV2920'DESCRIBED BY NATIONAL GEODETIC SURVEY 1978 KV2920'2.6 MI NW FROM SOMERVILLE. KV2920'2.6 MILES NORTHWEST ALONG STATE ROUTE 28 FROM THE COUNTY COURT HOUSE KV2920'AT SOMERVILLE, SET 40.0 FEET NORTHWEST OF THE ENTRANCE TO THE VILLA, KV2920'48.5 FEET SOUTHWEST OF THE CENTER LINE OF STATE ROUTE 28, 50.1 FEETKV2920'SOUTHEAST OF A FIRE HYDRANT AND 4.7 FEET EAST OF THE SOUTHWEST CORNER

KV2920'OF THE BRICK FOUNDATION OF THE SIGN WITH THE VILLA ON IT.

60

USEFUL FORMULAS FOR

COMPUTATION OF

DISTANCES AND DIRECTIONS ON SPCS

The Grid Scale Factor correction for projecting a curved line onto a plane

Measured Distance

EllipsoidDistanceH

N

h

GeoidEllip

soid

Topography

61

Grid Scale Factor K12for a Line from Point 1 to Point 2

Purpose: To correct for scale distortion due to the projection of the ellipsoid onto a plane.

Formula:

Where: K12-Grid Scale factor of a line between points 1 and 2. K1-Grid scale factor at point 1K2-Grid scale factor at point 2 Km-Grid scale factor at the line's mid- point.

K K 4K K612

1 m 2= + +

Grid Scale Factor K12for a Line from Point 1 to Point 2

Usage: A reasonable approximation for the above formula is to compute a simple average of K1 and K2 . A further approximation is to compute a single K value for the entire line or for the entire survey area. This will be demonstrated later in our traverse example.

62

Scale Factor for LCC (Aproximate)

Where:k - The Scale FactorN - Northing of the point

N0, r0, k0, ϕ0 - Zone Constants

PA North PA SouthN0 = 138 899.5967 N0 = 124 063.7931r0 = 6 375 155. r0 = 6 374 457.k0 = 0.9999 5684 k0 = 0.9999 5950ϕ0 = 41°.417 407 624 ϕ0 = 40 °.450 672 360

30

03

020

20

06

tan)(

2

)(

r

NN

r

NNkk

ϕ⋅−+

−+=

Where:k - The Scale FactorE - Easting of the pointE0, r0, k0 - Zone Constants

For New JerseyE0 = 150 000 k0 = 0.9999 = 1.23078 x10-14

or K = 0.9999 + (E - Eo)2 . 1.23.10-14

Scale Factor for TM (Aproximate)

20

200

21

)(r

EEkk −+=

202

1

r

63

Reducing Measured Horizontal Distance to Grid Distance

Purpose: To reduce a measured horizontal distance to the projection plane. As mentioned earlier, field measurements are carried out on the physical surface of the earth, while office computation are performed on the projection of the earth onto a plane. This reduction is in essence the bridge between field measurements and the computations on the state plane coordinate system.

S’

DTopographySea Level - Geoid

Sea Level - GeoidEllipsoidEllipsoid

HHhh NN

R

)NHR

R(DS'

++⋅=

)hR

R(DS'

+⋅=

Reducing Measured Horizontal Distance to Ellispoid

hRD

RS'

+=

64


Formula:

Where: S- Grid Distance D- Horizontal (Measured) Distance H- Mean Elevation (Above Mean Sea Level) N- Mean Geoid Height (About -32m in NJ) R- Mean Radius of the Earth (About 6,372,000m) K12-Grid Scale factor of the Line.

12K)NHR

R(DS

++=


Usage: Obtain the elevations of the terminal points of the line, the Geoid height of the region and compute.

65

Relationship Between Geodetic and Grid Azimuths

Purpose: To account for the convergence of the north direction towards the pole vs. parallel north direction on a plane

Formula: AZGrid = AZGeodetic - γ + (t-T)

Where: AZGrid- Grid AzimuthAZGeodetic- Geodetic Azimuthγ- Meridian Convergence(t-T)- Arc-to-chord correction

E

N

Ttt

α

δ

Projected Geodetic line

Grid NorthGeodetic North

Projected meridian

γ


1

2

Line of sight

Line of sight

66


Usage: This above formula (except for the t-T correction) has to be used only if we are given with Geodetic Azimuth and want to perform our computations on plane coordinates.

– Grid Azimuth is computed from an inverse – Geodetic Azimuth from NGS, GPS or Astronomical Az plus

Laplace Correcetion

Arc-to-Chord Correction (t-T) for line 1-2

Purpose: The (t-T) correction is due to the fact that the measured direction between two points is actually a curved line, on the surface on a body such as an ellipsoid, that passes through these points. When projected onto a plane, the geodetic direction looks like an arc not a straight line. The angle that we compute from field notes is defined by the difference between two measured directions. Thus, the computed angle differs from the plane angle that we have to use when working with the State Plane Coordinate System. This difference is expressed by (t-T).

67


Formula: (t-T)" = 25.4 × ∆N × ∆E × 10-10

Where:∆N = N2 - N1

∆E =

(t-T)" -Arc-to-chord correction in seconds of arc. N1-Northing of point 1 N2-Northing of point 2 E1-Easting of point 1 E2-Easting of point 2

E EE2 1

02−

−


Usage: The size of the correction is rather small and can be neglected for most ordinary work (not high accuracy). The sign of the correction is dependent on the direction of the line with respect to the north (Azimuth dependent). Table 3.1 presents the size (magnitude) of the (t-T) correction and figure 6, describes the sign convention for (t-T), as it applied for the NJ State Plane Coordinate System. One should note that the following example does not apply to States with Lambert Conformal Conic projection (e. g. PA) or states that their central meridian is not 150,000.

68

Table 3.1 Size of the correction in seconds of arc (“): (for NJ, Eo= 150,000)

7.6"5.1"2.5"0"20 km

3.8"2.5"1.3"0"10 km

1.9"1.3"0.6"0"5 km

0.8"0.5"0.3"0"2 km

0 300,000

50,000 250,000

100,000 200,000150,000∆N

Average Easting of the line

Sign of the correctionfor States using Transverse Mercator projection

Cen

tral

Mer

idia

n

++

++

--

--

WEST ∆E <Eo EAST ∆E >Eo

0° 0°

90°90°

180° 180°

270°270°

69

Sign of the correctionfor States using Transverse Mercator projection

Cen

tral

Mer

idia

n

WEST ∆E <Eo EAST ∆E >Eo

0° 0°

90°90°

180° 180°

270°270°

Observed Direction

Plane Direction

Table 3.1 Size of the correction in seconds of arc (“): (for LCC)

10.2"

5.1"

2.5"

1.0"

200,000

12.7"7.6"5.1"2.5"20 km

6.4"3.8"2.5"1.3"10 km

3.2"1.9"1.3"0.6"5 km

1.3"0.8"0.5"0.3"2 km

250,000150,000100,00050,000∆E

N-No

70

Sign of the correctionfor States using LCC projection

Central ParallelNORTH ∆N >No

SOUTH ∆N <No0°

90°

180°

270°

0°

90°

180°

270°

Observed Direction

Plane Direction

- +- +

++

--

Sign of the correctionfor States using LCC projection

Central ParallelNORTH ∆N >No

SOUTH ∆N <No0°

90°

180°

270°

0°

90°

180°

270°

Observed Direction

Plane Direction

71

10 steps for computing a traverse on NAD 83 SPCS

Computing a traverse on NAD 83 State Plane Coordinate System may include the following steps:

1. Obtain starting and ending coordinates and grid azimuth.

2. Compute preliminary Azimuth for each line.3. Compute preliminary coordinates for each traverse

point.

These three steps are essentially computing an open traverse without applying corrections for closure errors.


4. Obtain or compute approximate elevations (mean or for each point) of the traverse.Step 4 is necessary in order to reduce the distances from the topography to mean sea level. Unless high precision results are sought or there are substantial elevation differences between the various traverse points, a mean elevation of the area will be sufficient.

5. Compute grid scale factor (mean or individual for each line).Again, the choice between computing a mean value or individual scale factors for each line depends on the accuracy objectives of the project and the magnitude of elevation differences between the traverse points.

72


6. Reduce horizontal distances to grid distances.7. Compute (t-T) correction for each line (if necessary).

For most traverses, (t-T) correction is practically negligible. If the Northing component (usually called the Latitude) of a side of the traverse is less than 1 mile long, the maximum (t-T) correction will not exceed 1″ (second of arc). Thus, the correction is equal or smaller than the accuracy with which we are able to carry out our measurements. (t-T) becomes significant only for long Latitudes (larger than 10 km) and at the east/west most parts of the State. One can assume that the type of work for which (t-T) is significant will be done with GPS.


8. Apply (t-T) correction to each Azimuth (if necessary).

9. Balance (or adjust) the traverse.10. Compute final State Plane Coordinates for the

traverse points.Steps 9 and 10 are the standard procedures for balancing the traverse with the Compass Rule or adjusting it with Least Squares.

73

Traversing on SPCS

C2

1

B

A

D

Step 1: Obtain starting and ending coordinates and grid azimuth and place the information in a table to facilitate the computations.

Traversing on SPCSPoint Angle Distance AZIMUTH N E

A (m) (m) (m)

341° 22' 35.0"

B 60° 50' 37" 199694.411 130266.014

3581.556

1 295° 51' 32"

3092.569

2 161° 26' 40"

2906.883

C 185° 58' 52" 202121.962 124818.850

325° 30' 18.6"

D

Also given: Average Elevation of the traverse is: 200m above MSL

74

Traversing on SPCS

Point Angle Distance AZIMUTH N E A LAT DEP

341 22 35.0 B 60 50 37 199694.411 130266.014

3581.556 222 13 12.0 -2652.393 -2406.7311 295 51 32 197042.018 127859.283

3092.569 338 4 44.0 2868.973 -1154.5472 161 26 40 199910.991 126704.736

2906.883 319 31 24.0 2211.180 -1886.969C 185 58 52 202122.171 124817.767

Comp. 325 30 16.0D Given 325 30 18.6 202121.962 124818.850

Closure 0 0 2.6 -0.209 1.083

Steps 2 and 3: Compute preliminary Azimuth for each line and compute preliminary coordinates for each traverse points.

Traversing on SPCS

Step 4: Obtain or compute approximate elevations (mean or for each point) of the traverse.

The average elevation is given as 200m above MSL. It is assumed that there are no substantial height differences among the traverse points. Thus, the average value will be used.

75

Traversing on SPCS

Step 5: Compute Grid and Elevation scale factors (mean or individual for each line)

Point GSFB 0.9999048C 0.99990781 0.99990602 0.9999067

Mean 0.9999063

The Grid scale factor (GSF) is computed based on the approximate Easting of the points. The results of the interpolation are:

Traversing on SPCS

One should note that the largest error committed by using the mean GSF instead of GSF for individual points will occur on side B-1 (the longest). The magnitude of this error is about 3mm (0.01 ft). This error is smaller than our ability to measure the distance B-1 with a total station. Thus, it is justifiable to use a mean GSF for our example traverse.

76

Traversing on SPCSThe Elevation scale factor (ESF) is required in order to

reduce the traverse from the topography to the ellipsoid. The computation is based on the elevation of traverse points above MSL (actually the Geoid) and the height of the ellipsoid above the Geoid (N).

Finally, the combined scale factor = GSF×ESF = 0.99987994

Using the ratio:

Mean ESF is = 0.99997364

RR H N+ +

Data:R = 6372160H = 200N = -32

Traversing on SPCS

Step 6: Reduce horizontal distances to grid distances. The reduction of the distances is performed by multiplying each measured distance with the combined scale factor from step 5.

Line Dist Grid DistB-1 3581.56 3581.126

1-2 3092.57 3092.1982-C 2906.88 2906.534

77

Traversing on SPCS

Step 7: Compute (t-T) correction for each line.

Line ∆N ∆E (t-T)"

B-1 -2652.393 -20937.352 0.14

1-2 2868.973 -22717.991 -0.17

2-C 2211.180 -24238.749 -0.14

Traversing on SPCS

Step 8: Apply (t-T) correction to each Azimuth (if necessary).

It can be seen from step 7 that the magnitude of this correction is negligible for most projects. Only very precise traverses performed with first ordertheodolites may be subject to such a small correction. Nevertheless, let’s see how to apply the correction for reference purposes only.

78

Traversing on SPCS

Sincethe central meridian in NJ is 150,000 and our traverse points have smaller Eastings (between 124,000. and 131,000.), we have to use the west (left) side of the sign diagram

NW0°

NE

270° 90°

SW

180°

SE

+ +- -

Traversing on SPCS

From the preliminary traverse computation (steps 2-3) we have the following information:

Point Backsight AZ Quad./Sign Foresight AZ Quad./Sign

B 161 22 35.0 SE / - 222 13 12.0 SW / -

1 42 13 12.0 NE / + 338 4 44.0 NW / +

2 158 4 44.0 SE / - 319 31 24.0 NW / +

C 139 31 24.0 SE / - 325 30 16.0 NW / +

79

Traversing on SPCSCombining the sign information with the computed values for (t-T) the corrections for each observed angle are:

Observed BS FS FS-BS Corrected Point Angle Corr Corr Total Angle

B 60 50 37 0.0 -0.14 -0.14 60 50 36.91 295 51 32 0.14 0.17 0.03 295 51 32.02 161 26 40 -0.17 0.14 0.31 161 26 40.3C 185 58 52 -0.14 0.0 0.14 185 58 52.1

Note that the Backsight at point B and the foresight at point C are control points with State Plane coordinates. Thus, the Azimuths to these points are Grid Azimuths and thus already corrected for (t-T).

Traversing on SPCSStep 9: Balance (or adjust) the traverse. In this

example we use the Compass Rule. Balanced N E

Point Angle Distance Azimuth LAT DEPA

341 22 35.0 B 60 50 36.9 +0.6 199694.411 130266.014

3581.126 222 13 12.5 -2652.069 -2406.4481 295 51 32.0 +0.5 197042.342 127859.566

3092.198 338 4 45.0 2868.634 -1154.3942 161 26 40.3 +0.6 199910.975 126705.172

2906.534 319 31 25.9 2210.932 -1886.722C 185 58 52.1 +0.6 202121.907 124818.449

Closure 2.3” Comput. 325 30 18.6D Given 325 30 18.6 202121.962 124818.850

Closure 0 0 0 0.055 0.401Relative precision = 1: 23,698

80

Traversing on SPCS

Step 10: Compute final State Plane Coordinates for the traverse points.

Point N E

B 199694.411 130266.014

1 197042.362 127859.716

2 199911.014 126705.451

C 202121.962 124818.850

GPSing on SPCSA large tract of land has to be surveyed and delivered with State Plane Coordinates. The monuments of the two west corners of the tract were recovered and documents revealed that the Azimuth of the line is 29° 53’ (determined from “sun shots”). In order to tie the traverse to SPC, four GPS points were ordered from a GPS service firm.

GPS2GPS4

GPS3 GPS1

TR1

TR2 TR3

TR4

TR5TR6

81

GPSing on SPCS

The following data was obtained from the GPS observations:

Point E N Elev

GPS 1 106245.2422 131565.4149 89.45

GPS 2 106801.2074 132412.7078 67.91

GPS 3 101045.0197 133988.5688 101.43

GPS 4 101672.8785 135065.1054 117.18

GPSing on SPCS

You decided to check if the GPS data is correct by performing the following computations:

1. Compute the Bearing of the line from GPS3 to GPS4 and compare it with the given Azimuth. You discover that the computed Bearing is: 30°15’ 06” while the given Azimuth is 29° 53’. What is wrong?

(Answer: compute meridian convergence)

82

GPSing on SPCS

2. Now that you found a problem you decided to measure the distance between GPS3 and GPS4 with your EDM. The measured distance was 4088.356. The computed distance from GPS data is 1246.250. problem ?

You remember that NAD83 coordinates are in Meters, so you convert the measured distance to Meters. Now the computed distance is 1246.146. Why is there a 0.1m (0.3’) difference?

GPSing on SPCS

You realize that the discrepancy could be because of the Grid Scale Factor. Computing the corrected distance for GSF yields 1246.161. This improves the situation somewhat, but still the distances are about 1/4 ft off. How can you account for this difference?

(Answer: compute “sea level” SF)

83

Compute ground level distances from State Plane Coordinates.

In some cases it is necessary to compute actual ground level distances between points that have coordinates in SPCS. The actual distance is computed from:

DistanceDistance

actualgrid

Comb S F=

. . .

Compute ground level distances from State Plane Coordinates.

Thus, in order to compute the actual distance from a computed distance from coordinates we need to reverse the computation procedure we discussed earlier.

The reversed procedure is to compute GSF (Grid scale factor), ESF (elevation scale factor) and the combined scale factor as described in step 5. Subsequently, apply the correction by dividing the grid distance by the combined scale factor.

84

Compute ground level distances from State Plane Coordinates.For example, the actual ground distance between point 1 and 2 in our example is computed as follows:

(computed from coordinates)Grid Distance = 3092.166 m

Combined scale factor = 0.99987994

Actual ground distance = 3092.538 m

Computing Stakeout Distances from SPCS

85

Example

E N1 106245.242 131565.4142 106801.207 132412.707

3 101045.019 133988.5684 101672.878 135065.105

131000

131500

132000

132500

133000

133500

134000

134500

135000

135500

100000 102000 104000 106000 108000

1

4

3

2

Ground Distance Computation

Point 1 2E 106245.242 106801.207N 131565.414

1013.411132412.707

DistanceElevation 251.22 288.34

86


Point 1 2E 106245.242 106801.207N 131565.414

1013.411132412.707


Ground SF 0.9999235 0.9999230


Point 1 2E 106245.242 106801.207N 131565.414

1013.411132412.707


Ground SF 0.9999235 0.9999230Elevation SF 0.9999656 0.9999598

87


Point 1 2E 106245.242 106801.207N 131565.414

1013.411132412.707



Comb SF 0.99988915 0.99988273


Point 1 2E 106245.242 106801.207N 131565.414

1013.411132412.707



Comb SF 0.99988915-0.116

0.99988273SF Correction

1013.527SP Distance3325.1791Distance ft

88


Pt. E N Distance Elevation SP Distance

1 106245.242 131565.414 1013.411 251.22 1013.527

2 106801.207 132412.707 288.34 -0.116

3 101045.019 133988.568 1246.250 301.57 1246.373

4 101672.878 135065.105 117.18 -0.123

“Ground” Coordinates

106257.021 131580.000 1014.611 1013.527106813.733 132428.237 1.084101056.420 134003.687 1241.658 1246.373101681.484 135076.538 -4.715

Comp Dist Actual Dist

89

Shifted “Ground” CoordinatesShift to Lower Left Corner


5200.800 0.000 1013.548 1013.5275756.863 847.392 0.022

0.000 2423.427 1246.296 1246.373627.912 3499.987 -0.077

Relative “Ground” CoordinatesShift to Project Center


2293.332 -1706.891 1013.529 1013.5272849.376 -859.509 0.002

-2907.474 716.533 1246.379 1246.373-2279.480 1793.141 0.006

90


52293.332 78293.109 1013.529 1013.52752849.376 79140.491 0.00247092.526 80716.533 1246.379 1246.37347720.520 81793.141 0.006

Shifted+Relative “Ground” Coordinates(Project Center +50,000, 80,000)

The End

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