1. Trigonometric heighting.
Surveying I. (BSc)
Lecture 5.
Trigonometric heighting.
Distance measurements, corrections and reductions

2. How could the height of skyscrapers be measured?

3. The principle of trigonometric heighting

4. The principle of trigonometric heighting

5. The principle of trigonometric heighting

6. The principle of trigonometric heighting

7. The principle of trigonometric heighting

8. Trigonometric levelling

9. Trigonometric levelling

10. Trigonometric levelling

11. Trigonometric levelling

12. Trigonometric levelling

13. Trigonometric levelling

14. Trigonometric levelling
Advantage:
the instrument height is not necessary;
non intervisible points can be measured, too.

15. 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

16. The determination of the heights of buildings

17. The determination of the heights of buildings

18. The determination of the heights of buildings

19. The determination of the heights of buildings
The horizontal distance is observable, therefore:

20. Determination of the height of buildings
The distance is not observable.

21. Determination of the height of buildings

22. Determination of the height of buildings

23. Determination of the height of buildings

24. Determination of the height of buildings

25. Determination of the height of buildings

26. Determination of the height of buildings
Using the sine-theorem:

27. Determination of the height of buildings

28. Determination of the height of buildings

29. Determination of the height of buildings
Using the observations in pont B:

30. Trigonometric heighting The effect of Earth’s curvature

31. Trigonometric heighting The effect of Earth’s curvature

32. Trigonometric heighting The effect of Earth’s curvature
The central angle:

33. Trigonometric heighting The effect of Earth’s curvature
The tangent-chord angle is equal to g/2.

34. Trigonometric heighting The effect of Earth’s curvature

35. Trigonometric Heighting The effect of refraction

36. Trigonometric Heighting The effect of refraction

37. Trigonometric heighting The effect of refraction
Let’s introduce the refractive coefficient:
Thus Dm can be computed:
where:

38. Trigonometric heighting
The combined effect of curvature and refraction
Note that the effects have opposite signs!

39. =Dr Trigonometric heighting
The combined effect of curvature and refraction
=Dr
The elevation difference between A and B (the combined effect of curvature and refraction is taken into consideration):
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.

40. Determination of distances
Distance: is the length of the shortest path between the points
projected to the reference level

41. 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)

42. 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.

43. Reduction of slope distance to the horizontal plane
The slope distance is measured along the terrain
Suppose that the angle (ai) between the li distance and the horizon is known, thus
or:
where:
and:

44. Reduction of slope distance to the horizontal plane
The slope distance is measured between the points directly.
When the elevation difference is known:
where:

45. Determination of distance on the reference surface
Reduction of horizontal distance to the reference level (MSL)
Thus the distance on the reference surface:
The reduction is:

46. 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).

47. 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. . , ,

48. 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:
The difference of the true length and baseline length from N number
of repeated observations:

49. Corrections of the length observations
Standardization correction (takes into account the difference between the nominal and the true length):
where l is the standardized length and (l) is the nominal length
Temperature correction (takes into consideration the thermal expansion of the tape):
(steel)
Thus the corrected length: