Surveying I. (BSc)
Lecture 5.
Trigonometric heighting.
Distance measurements, corrections and
reductions
How could the height of skyscrapers be measured?
?
?
The principle of trigonometric heighting
The principle of trigonometric heighting
The principle of trigonometric heighting
The principle of trigonometric heighting
The principle of trigonometric heighting
m  h  m    h    d cot z
Trigonometric levelling
Trigonometric levelling
Trigonometric levelling
Trigonometric levelling
Trigonometric levelling
Trigonometric levelling
Trigonometric levelling
Advantage:
• the instrument height is not
necessary;
• non intervisible points can be
measured, too.
m  d B cot z B   B   d A cot z A   A  
 t B cos z B   B   t A cos z A   A 
Trigonometric heighting
Advantages compared to optical levelling:
• A large elevation difference can be measured over short
distances;
• The elevation difference of distant points can be measured
(mountain peaks);
• The elevation of inaccessible points can be measured (towers,
chimneys, etc.)
Disadvantages 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
The determination of the heights of buildings
The determination of the heights of buildings
The determination of the heights of buildings
The determination of the heights of buildings
The horizontal distance is observable, therefore:
m  d AP cot z A
m  lO  d AP cot z A
Determination of the height of buildings
The distance is not observable.
Determination of the height of buildings
Determination of the height of buildings
Determination of the height of buildings
Determination of the height of buildings
Determination of the height of buildings
Determination of the height of buildings
Using the sine-theorem:
d AP
a
sin 

 d AP  a
sin  sin 180    
sin    
d BP
a
sin

 d BP  a
sin sin180     
sin   
Determination of the height of buildings
Determination of the height of buildings
m  lO  d AP cot z A
A
Determination of the height of buildings
Using the observations in pont B:
m  l  d BP cot zB
B
B
O

m
m
A
m
2
B

Trigonometric heighting
The effect of Earth’s curvature
Trigonometric heighting
The effect of Earth’s curvature
Trigonometric heighting
The effect of Earth’s curvature
The central angle:
d AB

R
Trigonometric heighting
The effect of Earth’s curvature
The tangent-chord angle is equal to /2.
Trigonometric heighting
The effect of Earth’s curvature
The effect of Earth’s curvature:

2
d AB d AB
 sz  d AB  tan  d AB 

2
2R
2R
Trigonometric Heighting
The effect of refraction
Trigonometric Heighting
The effect of refraction
d AB

2
m  d  cotzAB   
m  d  cot zAB  d  
2
d
m  d  cot zAB 
2
Trigonometric heighting
The effect of refraction
Let’s introduce the refractive coefficient:
k
R

 0,13
Thus m can be computed:
2
d
m  d  cot zAB 
 d  cot zAB   r
2
where:
d2
d2
r 
k
2
2R
Trigonometric heighting
The combined effect of curvature and refraction
Note that the effects have
opposite signs!
Trigonometric heighting
The combined effect of curvature and refraction
2
d
m  d  cot zAB  k
2R
=r
d2
 sz 
2R
The elevation difference between A and B (the combined effect of
curvature and refraction is taken into consideration):
d2
m  h  l  d  cot zAB  1  k 
2R
The fundamental equation of trigonometric heighting
The combined effect reaches the level of 1 cm in the distance of
d  0,4 km = 400 m.
Determination of distances
Distance: is the length of the shortest path between the points
projected to the reference level
Determination of distances
Distance: is the length of the shortest path between the points
projected to the reference level
The distance at the reference level
can not be observed, therefore the
slope distance must be measured in
any of the following ways:
• It can be the shortest distance
between the points (t)
Determination of distances
Distance: is the length of the shortest path between the points
projected to the reference level
The distance at the reference level
can not be observed, therefore the
slope distance must be measured in
any of the following ways:
• It can be the shortest distance
between the points (t)
• The distance measured
along the intersection of the
vertical plane fitted to A and
B, and the surface of the
topography.
Reduction of slope distance to the horizontal plane
The slope distance is measured along the terrain
Suppose that the angle (i)
between the li distance and the
horizon is known, thus
 v,i   i cosi
or:
mi

2 i
 v,i   i  v,i
where:
and:
mi2
 v ,i  
2 i
tv   lv,i
Reduction of slope distance to the horizontal plane
The slope distance is measured between the points directly.
tv  t f sin z
When the elevation difference is
known:
tv  t f  v
where:
m
v  
2t f
2
Determination of distance on the reference surface
Reduction of horizontal distance to the reference level (MSL)
Thus the distance on the
tg
reference surface:
R

tv RH H
t g  tv  tv
 tv   v
R
RH H
H

The reduction
is:  1 
tv
RH
RH
tg
H
v 
  tv
R
tg
H
 1
tv
R
Determination of distances
Distances can be measured directly, when a tool with a given
length is compared with the distance (tape, rod, etc.)
Distances can be measured indirectly, when geometrical of
physical quantities are measured, which are the function of the
distance (optical or electronical methods).
Standardization of the tape
How long is a tape in reality?
The length of the tape depends on
• the tension of the tape, therefore tapes must be pulled with the standard
force of 100 N during the observation and the standardization;
.
,,
• the temperature of the tape, therefore the temperature must be measured
during observations (tm) and during the standardization (tk), too.
Standardization of the tape
The real length of the tape
The difference between the true length (l) and the baseline length (a)
from a single observation:
l
r
The difference of the true length and baseline length from N number
of repeated observations:
i
  a  
d d d
d
 
n
Corrections of the length observations
Standardization correction (takes into account the difference
between the nominal and the true length):
 k    
where l is the standardized length and (l) is the nominal length
Temperature correction (takes into consideration the thermal
expansion of the tape):
t   tm  tk 
Thus the corrected length:
     k   t
  1,1105 / C
(steel)