Surveying and Surveying and GeometryGeometry

Brittany Crawford-PurcellBrittany Crawford-Purcell

What is Surveying?What is Surveying?

►Science of accurately determining the Science of accurately determining the terrestrial or three-dimensional terrestrial or three-dimensional position of points and the distances position of points and the distances and angles between them. and angles between them.

Prolong a Straight Line Forward Prolong a Straight Line Forward from an Existing Point from an Existing Point

Prolong a Straight Line Forward Prolong a Straight Line Forward from an Existing Pointfrom an Existing Point

Line Needs to Extending Line Needs to Extending Through an ObstructionThrough an Obstruction

► 1. Find 1. Find appropriate point appropriate point C at angle α from C at angle α from AB direction. AB direction.

► 2. Turn angle -2α 2. Turn angle -2α at C and locate at C and locate point D such that point D such that CD = BC. CD = BC.

► 3. Turn angle α at 3. Turn angle α at D to locate E and D to locate E and extension of extension of original line. original line.

The Collinearity of A, B, D The Collinearity of A, B, D and Eand E

The line AB extended through B must meet CD say at some point D'. 

The Collinearity of A, B, D The Collinearity of A, B, D and Eand 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

The Collinearity of A, B, D The Collinearity of A, B, D and Eand E

Becasuse <CBD’ and <CD’B are equal = α

CD= BC = CD’, D=D’

A, B, D are collinear

Horizontal Distance of a Horizontal Distance of a Surface Surface

► A map is flat and shows all the points A map is flat and shows all the points on the same levelon the same level

►But the surface of the earth is rarely But the surface of the earth is rarely flat due to all the local ups and downs flat due to all the local ups and downs

►How do you calculate the distance How do you calculate the distance between two objects of different between two objects of different height?height? Use the distance between two objects (on Use the distance between two objects (on

the slope) and the correction term Cthe slope) and the correction term Chh

Horizontal Distance of a Horizontal Distance of a Surface 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

Horizontal Distance of a Horizontal Distance of a SurfaceSurface

cos α = d/L d=L* cos α

Ch= L- d = L- (L* cos α)Ch= L (1- cos α)