When calculating the correction for curvature in levelling over a distance D, which of the following formulas accurately represents the correction C in meters for the curvature effect, assuming the Earth is a perfect sphere with a radius R?
- AC = D²/R
- BC = D²/2RCorrect
- CC = D²/4R
- DC = D²/8R
Solution
Concept
Earth curvature correction formula in levelling
Formula
C = D2 / (2R)
Step-by-step solution
1. The curvature correction accounts for the departure of the Earth's spherical surface from a horizontal tangent plane over a distance D. 2. Assuming the Earth is a sphere of radius R, the formula for curvature correction C in meters is given by C = D²/8R. Answer: B
Explanation
Due to the spherical surface of the Earth, the horizontal line of sight diverges from the level surface over long sight distances. Based on geometry and the Pythagorean theorem for a circle of radius R, the vertical displacement due to Earth's curvature is calculated as C equals D squared divided by 2R. Consequently, option B gives the mathematically correct formula for curvature correction.
Why the other options are wrong
A) D²/R does not incorporate the factor required for curvature correction. C) D²/4R is not the correct expression for curvature correction.
Shortcut
Curvature correction formula: C = D² / 2R.
Common mistake
Selecting option A instead of B by forgetting the factor of 2 in the denominator of the curvature correction formula C = D²/2R.
Correct answer: B) C = D²/2R