Surveying I BSc Lecture 5 Trigonometric heighting Distance

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

Trigonometric levelling

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

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 horizontal distance is observable, therefore:

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

Determination of the height of buildings

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

Determination of the height of buildings

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

Trigonometric heighting The effect of Earth’s curvature

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

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

Trigonometric heighting The effect of Earth’s curvature:

$Trigonometric Heighting The effect of refraction$

Trigonometric Heighting The effect of refraction

$Trigonometric Heighting The effect of refraction$

Trigonometric Heighting The effect of refraction

$Trigonometric heighting The effect of refraction Let’s introduce the refractive coefficient: Thus Dm can$

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

$Trigonometric heighting The combined effect of curvature and refraction Note that the effects have$

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

$Trigonometric heighting The combined effect of curvature and refraction =Dr The elevation difference between$

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.

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 (ai) between the li distance and the horizon is known, thus or: where: and:

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

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:

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

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 thermal expansion of the tape): (steel) Thus the corrected length: