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

Jens Flemming

TL;DR

This work addresses the problem of computing the smallest enclosing circle on a sphere for a set of $n$ points, including a key hemisphere containment scenario. It adapts Welzl's linear-time algorithm for the 3D smallest enclosing sphere, augmented with a real-time hemisphere detector, to achieve linear expected runtime when the data lie in a hemisphere, and mirrors a 2D Welzl variant in that case. A rigorous derivation shows how boundary-size constraints and specific states enable hemisphere versus full-sphere detection, with implementation details such as plane-based containment tests and a move-to-front structure. For the full-sphere case, the paper notes that linear-time solutions are not known and points to spherical Voronoi-based approaches with $O(n\log n)$ complexity as a practical fallback, accompanied by a ready-to-use Python implementation on GitHub.

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.

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

TL;DR

This work addresses the problem of computing the smallest enclosing circle on a sphere for a set of points, including a key hemisphere containment scenario. It adapts Welzl's linear-time algorithm for the 3D smallest enclosing sphere, augmented with a real-time hemisphere detector, to achieve linear expected runtime when the data lie in a hemisphere, and mirrors a 2D Welzl variant in that case. A rigorous derivation shows how boundary-size constraints and specific states enable hemisphere versus full-sphere detection, with implementation details such as plane-based containment tests and a move-to-front structure. For the full-sphere case, the paper notes that linear-time solutions are not known and points to spherical Voronoi-based approaches with complexity as a practical fallback, accompanied by a ready-to-use Python implementation on GitHub.

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.
Paper Structure (10 sections, 4 theorems, 4 equations, 1 figure, 2 algorithms)

This paper contains 10 sections, 4 theorems, 4 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Problem statement and applications
  2. Algorithms in the literature
  3. The algorithm
  4. Welzl's algorithm for planar circles and spheres
  5. Smallest spherical circles vs. smallest spheres
  6. Derivation of the algorithm
  7. Statement of the algorithm
  8. Implementation details
  9. The full-sphere case
  10. Conclusions

Key Result

Lemma 1

The small cap defined by a spherical circle with radius $r<\frac{\pi}{2}$ and center $c$ equals the intersection of the sphere and the ball with radius $\sin c$ centered at $(\cos r)\,c$.

Figures (1)

  • Figure 1: Runtime for point clouds of different sizes between 100 000 and 1 million points. Points are uniformly distributed in a 'rectangle'. Longitudes span 90 degrees, latitudes span 60 degrees. For each cloud size the algorithm is run 10 times. Computations were done on one core of an Intel Core i7-8550U CPU with 1.8 GHz. Code is available in github.

Theorems & Definitions (10)

  • Definition 1
  • Lemma 1
  • proof
  • Definition 2
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • proof