Quadratic-Order Geodesics on Meshes

TL;DR

This paper presents a new quadratic-order method for computing discrete geodesics on triangle meshes, improving accuracy and artifact reduction over existing linear methods, especially on coarse or uneven meshes.

Contribution

It introduces a novel quadratic representation and optimization framework for geodesics that enhances accuracy and flexibility compared to prior linear approaches.

Findings

Exact reproduction of flat distances regardless of mesh quality

Improved accuracy on curved meshes

Supports sources anywhere on the mesh surface

Abstract

We introduce a novel representation and optimization framework for discrete geodesics on triangle meshes that reduces artifacts of linear methods on uneven and coarse discretizations. Our method computes squared geodesic distances from point and curve sources using piecewise-quadratic elements, exactly reproducing flat distances regardless of mesh quality while improving accuracy over existing approaches on curved meshes. The formulation naturally supports sources placed anywhere on the mesh, not just at vertices.