4.lecture trig level 1
TRANSCRIPT
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Trigonometric heighting.Distance measurements, corrections and
reductions
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Trigonometric Leveling
It is the branch of leveling in which therelative elevations of dierent stationsare determined from the observedvertical angles and known distances.
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V = S Sin α = H Tan α
ZB = Z
A + hi + S Sin α – r
ZB = Z
A + hi + H Tan α – r
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How could the height of skyscrapers be measured?
? ?
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The principle of trigonometric heighting
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The principle of trigonometric heighting
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The principle of trigonometric heighting
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The principle of trigonometric heighting
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The principle of trigonometric heighting
z d hmhm cot+−=−∆+=
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Trigonometric levelling
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Trigonometric levelling
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Trigonometric levelling
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Trigonometric levelling
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Trigonometric levelling
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Trigonometric levelling
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Trigonometric levelling
( ) ( )
( ) ( ) A A A B B B
A A A B B B
z t z t
z d z d m
−−−=
=−−−=
coscos
cotcot
Advantage:
• the instrument height is notnecessary;• non intervisible points can be
measured, too.
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Trigonometric heighting
Advantages compared to optical levelling:
• A large elevation difference can be measured over shortdistances;
• The elevation difference of distant points can be measured(mountain peaks);
• The elevation of inaccessible points can be measured (towerschimneys etc!)
"isadvantages compared to optical levelling:
• The accuracy of the measured elevation difference is usually
lower!
• The distance between the points must be known (or measured) in
order to compute the elevation difference
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The determination of the heights of buildings
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The determination of the heights of buildings
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The determination of the heights of buildings
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The determination of the heights of buildings
The horizontal distance is observable, therefore:
A AP z d m cot=∆
A AP O z d l m cot+=
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"etermination of the height of buildings
The distance is not observable.
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"etermination of the height of buildings
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"etermination of the height of buildings
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"etermination of the height of buildings
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"etermination of the height of buildings
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"etermination of the height of buildings
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"etermination of the height of buildings
Using the sine-theorem:
( ) ( )β α β
β α β +=⇒
−−=
sin
sin
180sinsinad
ad AP
AP
( ) ( )β α
α
β α α +=⇒
−−=
sin
sin
180sinsinad
ad
BP
BP
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"etermination of the height of buildings
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"etermination of the height of buildings
A AP O
A z d l m cot+=
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"etermination of the height of buildings
Using the observations in pont B:
B BP BO
B z d l m cot+=
( )
B Amm
m +
=
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• When long distances are involved it isessential to consider the eect of Curvature of Earth and Refractiondue to atmospheric conditions.
• The eect of curvature is to make theobjects appear lower than they really are.
• The eect of refraction is to make them
appear higher than they really are. It istaken as one sixth of that of curvature.
• The combined eect is to cause the objectappear lower than they really are.
Trigonometric Leveling
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Trigonometric heightingThe effect of #arth$s curvature
i i i i
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Trigonometric heightingThe effect of #arth$s curvature
T i i h i h i
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Trigonometric heightingThe effect of #arth$s curvature
R
d AB=
γ
The central angle:
T i t i h i hti
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Trigonometric heightingThe effect of #arth$s curvature
The tangent-chord angle is eual to γ !".
T i t i h i hti
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Trigonometric heightingThe effect of #arth$s curvature
The effect of #arth$s curvature:
Rd
Rd d d AB AB
AB AB sz
tan
=⋅≈⋅=∆ γ
T i t i H i hti
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Trigonometric HeightingThe effect of refraction
T i t i H i hti
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Trigonometric HeightingThe effect of refraction
ρ δ
ABd
=
ρ
cot
d
z d m AB −′⋅≈∆
T i t i h i hti
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Trigonometric heightingThe effect of refraction
%et$s introduce the refractive coefficient:
ρ
Rk =
Thus m can be computed:
r AB AB z d
d z d m ∆−′⋅=−′⋅≈∆ cot
cot
ρ
where:
R
d k
d r
==∆ ρ
T i t i h i hti
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Trigonometric heightingThe combined effect of curvature and refraction
&ote that the effects haveopposite signs'
T i t i h i hti
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Trigonometric heightingThe combined effect of curvature and refraction
R
d k z d m AB
cot
−′⋅=∆
R
d sz
=∆
=∆r
( ) R
d k z d hm
AB
1cot
−+′⋅+−= l
The elevation difference between A and (the combined effect of
curvature and refraction is taken into consideration):
The fundamental equation of trigonometric heighting
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• #$$ect o$ %urvature
• % & '.'"() * suare + ''' o$ $t
• or
• % & '.'/01 suare 2$ distance in m• #$$ect o$ 3e$raction
• 3& '.''(( * suare + ''' o$ $t
• or
• 3 & '.'1 suare +2$ distance in m• %ombined #$$ect o$ % and 3
• 4 & '.'"'5 * suare + ''' o$ $t
• or
• 4 & '.'501 suare 2$ distance in m
Constants