4.lecture trig level 1

8/19/2019 4.Lecture Trig Level 1

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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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z d hmhm cot+−=−∆+=


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Trigonometric levelling


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( ) ( )

( ) ( ) 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 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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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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A AP O

A z d l m cot+=


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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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T i i h i h i


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R

d AB=

γ

The central angle:

T i t i h i hti


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The tangent-chord angle is eual to γ !".

T i t i h i hti


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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

4.lecture trig level 1

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