Intrinsic Meshing of Closed Surfaces Using Geodesic Distances

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.