Slope and Altitude

When a tape is suspended between two measuring heads . And B   both at same level , the shape it takes up is a catenary.

 if C is the lowest point on the curve , then on length CB there are three forces acting , namely the tension T at B to at C and the weight of portion CB where is the weight per unit length and s is the are length CB thus must be in equilibrium under the action of these three forces hence.

Slope:

if the difference in height of the two measuring heads is h , the slope distance L andhorizontal equivalent D then by pythagoras:

D  =  (L ² - h ²) 1/2

Alternatively if the vertical angle of the slope of the ground is measured then:

D = L cos ⦵

and the correction C ⦵  = L - D 

C ⦵ = L ( 1 - cos ⦵ )

Altitude:

if the surveys are to be connected to the national mapping system of country , the distance will need to be reduces to the common datum of that system , namely MSL Alternatively if the engineering scheme is of a local nature. 

distances may be reduced to the mean level of the area . This has the advantage setting -out distances on the ground are .  without sensible error equal to distance computed from coordinates in the mean datum plane.

Consider in whch a distanced in a plane situated at a height H above MSL.

                                              R

By similar triangles   M =  ___________  ×  L

                                       R + H

                                                           RL                                  R                           LH

∴  Correction  Cm = L - M  = L  -  _______   =  L  ( 1 -  _________  )  =  ___________

                                                           R+H                            R + H                    R + H

As H is normally negligible compared with R in the denominator 

                LH

CM  =  ______

                  R

The correction is negative for surface work above MSL or mean level of the area but may be positive for tunnelling or mining work below MSL or mean level of the area.