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

TL;DR

This paper introduces a simple linear time algorithm for finding the smallest enclosing circle on a sphere, which automatically detects hemisphere conditions and extends to full-sphere cases with suggested adaptations.

Contribution

It presents a novel linear time algorithm for smallest enclosing circles on a sphere, improving efficiency and handling hemisphere detection automatically.

Findings

Algorithm runs in linear time for hemisphere-contained point clouds.

Automatically detects whether points are contained in a hemisphere.

Provides guidance for adapting to full-sphere cases.

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.