Comparative analysis of the ellipsoid approximation with the sphere

Abstract

The paper analyzes the approximation of the ellipsoid by the sphere. Earth is a space body with a mathematically irregular shape, so idealized smooth surfaces are used for calculations. The first is the geoid, a smooth, equipotential surface that best approximates mean sea level. However, the geoid does not have an analytical form and is unsuitable for many applications, so an ellipsoid is used for approximation. In applications where high accuracy is not required (e.g., with small scale maps), the ellipsoid is approximated by a sphere. The radius of the sphere can be calculated in three ways: according to the equivalent volume criterion, according to the equivalent surface criterion, or as the mean value of the three semi-axes of the ellipsoid. All three methods of approximation were tested by calculating the length of the geodetic line on the ellipsoid, the orthodrome on the spheres and then the error. Also, the influence of latitude on the error value was tested. For three different values of geographic latitude, the lengths of geodetic lines up to one hundred points were calculated (using the Bessel method for solving the second main geodetic task on the ellipsoid), as well as the lengths of the orthodromes on the spheres, with the radii of the spheres determined in the three mentioned ways. The obtained results were then analyzed and discussed.

Keywords

• orthodrome
• loxodrome
• geodetic line
• approximation
• analysis