Geodesics in Heat

TL;DR

The paper presents the heat method, a robust and efficient approach for computing geodesic distances on various geometric domains using standard linear elliptic problems, significantly improving speed over existing methods.

Contribution

It introduces the heat method for geodesic distance computation, enabling fast, accurate, and easy implementation with precomputable linear systems.

Findings

Faster distance updates compared to state-of-the-art methods

Linear systems can be prefactored for near-linear time solutions

Convergence to exact geodesic distance demonstrated

Abstract

We introduce the heat method for computing the shortest geodesic distance to a specified subset (e.g., point or curve) of a given domain. The heat method is robust, efficient, and simple to implement since it is based on solving a pair of standard linear elliptic problems. The method represents a significant breakthrough in the practical computation of distance on a wide variety of geometric domains, since the resulting linear systems can be prefactored once and subsequently solved in near-linear time. In practice, distance can be updated via the heat method an order of magnitude faster than with state-of-the-art methods while maintaining a comparable level of accuracy. We provide numerical evidence that the method converges to the exact geodesic distance in the limit of refinement; we also explore smoothed approximations of distance suitable for applications where more regularity is required.