Intrinsic Meshing of Closed Surfaces Using Geodesic Distances

2026-07-06

AI Summary

This work proposes a novel method for constructing high-quality intrinsic triangulations of closed discrete surfaces without altering their original geometry. By performing local operations—such as edge flips, splits, and collapses—directly on the surface and integrating the continuous Dijkstra algorithm with an A* search acceleration strategy, the approach efficiently computes exact geodesic distances, overcoming the traditional reliance on developable triangles. The method introduces an angle quality criterion based on intrinsic distances and employs a feature-length field to control element sizing, enabling adaptive mesh refinement and coarsening, thereby laying the groundwork for high-order mesh generation. Experiments on nearly 5,000 complex models from the Thingi10K dataset demonstrate that the computational cost of geodesic distance calculation is reduced to approximately 3% of that required by conventional methods.

Abstract

We present a method for constructing intrinsic triangulations of closed discrete surfaces, in which edges correspond to shortest geodesic paths and faces decompose into geometric primitives inherited from the underlying mesh. Starting from a watertight input triangulation, the method progressively builds an intrinsic mesh through local optimization operations -- edge swaps, edge splits, edge collapses, and triangle splits -- performed directly on the surface without modifying the original geometry. Element size is controlled via a characteristic length field, and quality is enforced through angle-based criteria derived from intrinsic distances. Geodesic distances are computed exactly using a continuous Dijkstra approach, accelerated by an A* search strategy that reduces computation to roughly $3\%$ of the cost of standard propagation. The framework supports both refinement and coarsening, overcoming a key limitation of prior intrinsic methods based on developable triangles. As a by-product, the intrinsic triangulation provides a natural foundation for direct high-order mesh generation, bypassing the classical pipeline of first constructing a linear mesh and subsequently curving it. The method is validated on the Thingi10K dataset across nearly 5,000 geometrically complex models.

Problem

• intrinsic meshing
• geodesic distances
• closed surfaces
• triangulation
• mesh refinement

Innovation

• intrinsic triangulation
• geodesic distance
• mesh optimization
• high-order mesh generation
• A* accelerated Dijkstra