Approximate calculation of Crofton formula

Abstract

Crofton formula has attracted much attention in differential geometry, which can be used to calculate the length of the curve. This article starts with the concept and definition of Crofton formula on the sphere, and gives out approximate calculation formula of a curve on spherical surface based on surf segmentation. The triangular grid of the geodesic dome and Marsaglia Polar Method are adopted to generate random points for an equal division of the sphere. This article takes the circular arc on the sphere as an example, through MATLAB simulation, the length was calculated by approximation of Crofton’s formula. This calculation method can be extended to approximate calculation of the length of any curve in space, which will have a profound impact on practical engineering applications.