A simple linear time algorithm for smallest enclosing circles on the (hemi)sphere

Abstract

Based on Welzl's algorithm for smallest circles and spheres we develop a simple linear time algorithm for finding the smallest circle enclosing a point cloud on a sphere. The algorithm yields correct results as long as the point cloud is contained in a hemisphere, but the hemisphere does not have to be known in advance and the algorithm automatically detects whether the hemisphere assumption is met. For the full-sphere case, that is, if the point cloud is not contained in a hemisphere, we provide hints on how to adapt existing linearithmic time algorithms for spherical Voronoi diagrams to find the smallest enclosing circle.