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Intrinsic Delaunay triangulation (IDT) is a fundamental data structure incomputational geometry and computer graphics. However, except for sometheoretical results, such as existence and uniqueness, little progress has beenmade towards computing IDT on simplicial surfaces. To date the only way forconstructing IDTs is the edge-flipping algorithm, which iteratively flips thenon-Delaunay edge to be locally Delaunay. Although the algorithm isconceptually simple and guarantees to stop in finite steps, it has no knowntime complexity. Moreover, the edge-flipping algorithm may produce non-regulartriangulations, which contain self-loops and/or faces with only two edges. Inthis paper, we propose a new method for constructing IDT on manifold trianglemeshes. Based on the duality of geodesic Voronoi diagrams, our method canguarantee the resultant IDTs are regular. Our method has a theoreticalworst-case time complexity $O(n^2\log n)$ for a mesh with $n$ vertices. Weobserve that most real-world models are far from their Delaunay triangulations,thus, the edge-flipping algorithm takes many iterations to fix the non-Delaunayedges. In contrast, our method is non-iterative and insensitive to the numberof non-Delaunay edges. Empirically, it runs in linear time $O(n)$ on real-worldmodels.

Constructing Intrinsic Delaunay Triangulations from the Dual of Geodesic Voronoi Diagrams