Quadratic-Order Geodesics on Meshes

2026-03-03

AI Summary

Existing linear methods for computing discrete geodesic distances often suffer from artifacts and insufficient accuracy on non-uniform or coarse meshes. This work proposes a geodesic distance representation and optimization framework based on piecewise quadratic finite elements, which, for the first time, employs a second-order representation to exactly reproduce geodesic distances in flat regions without relying on mesh quality. The approach naturally accommodates point or curve sources at arbitrary locations—including non-vertex positions—and seamlessly integrates discrete differential geometry with geodesic distance optimization. Extensive evaluations demonstrate that the method significantly enhances both accuracy and robustness across a variety of triangular meshes, particularly excelling on coarse or highly non-uniform discretizations.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• geodesics
• triangle meshes
• discrete geodesics
• mesh discretization
• squared geodesic distances

Innovation

• quadratic geodesics
• mesh discretization
• squared geodesic distance
• piecewise-quadratic elements
• discrete geodesics