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Surveying and Geometry

Surveying and Geometry. Brittany Crawford-Purcell. What is Surveying?. Science of accurately determining the terrestrial or three-dimensional position of points and the distances and angles between them. Prolong a Straight Line Forward from an Existing Point.

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Surveying and Geometry

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  1. Surveying and Geometry Brittany Crawford-Purcell

  2. What is Surveying? • Science of accurately determining the terrestrial or three-dimensional position of points and the distances and angles between them.

  3. Prolong a Straight Line Forward from an Existing Point

  4. Prolong a Straight Line Forward from an Existing Point

  5. Line Needs to Extending Through an Obstruction • 1. Find appropriate point C at angle α from AB direction. • 2. Turn angle -2α at C and locate point D such that CD = BC. • 3. Turn angle α at D to locate E and extension of original line.

  6. The Collinearity of A, B, D and E The line AB extended through B must meet CD say at some point D'. 

  7. The Collinearity of A, B, D and E In a triangle each exterior angle is equal to the sum of the other two interior angles. Therefore <CBD’ and <CD’B are equal = α

  8. The Collinearity of A, B, D and E Becasuse <CBD’ and <CD’B are equal = α CD= BC = CD’, D=D’ A, B, D are collinear

  9. Horizontal Distance of a Surface • A map is flat and shows all the points on the same level • But the surface of the earth is rarely flat due to all the local ups and downs • How do you calculate the distance between two objects of different height? • Use the distance between two objects (on the slope) and the correction term Ch

  10. Horizontal Distance of a Surface Ch= L- d =L - √(L2-h2) Using the Pythagorean Theorem =L - L(1 - (h/L)2)1/2 Newton's binomial expansion (1 -x)1/2 = 1 - x/2 - x2/8 + ... with x = (h/L)2 Ch = h2/2L + h 4/ 8L3

  11. Horizontal Distance of a Surface cos α = d/L d=L* cos α Ch= L- d = L- (L* cos α) Ch= L (1- cos α)

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