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You Can Find Geodesic Paths in Triangle Meshes by Just Flipping Edges
Event Type
Technical Papers
Technical Papers Q&A
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TimeSunday, 13 December 202012:12 - 12:18 SGT
LocationZoom Room 1
DescriptionThis paper describes a new approach to computing exact geodesics on polyhedral surfaces. The basic idea is to perform edge flips, in the same spirit as the classic Delaunay flip algorithm. As a natural byproduct of this process, one also obtains a triangulation containing the shortened paths as edges. More precisely, given a path as a sequence of mesh edges, we transform it into a locally shortest geodesic path while avoiding self-crossings (i.e., we find a geodesic in the same isotopy class). Implementation amounts to a simple subroutine that flips edges to create a shorter path within their local neighborhood. The method is guaranteed to produce an exact geodesic path in a finite number of operations; practical runtimes are on the order of a few milliseconds, even for meshes with millions of faces. It is easily applied to cases beyond simple paths, including closed loops, curve networks, and multiply-covered curves. The method is well-suited for applications in geometry processing: it guarantees that curves do not cross, a necessary property when straightening region boundaries or cut networks, and furthermore it yields an intrinsic triangulation which includes the geodesic curves as edges, extending the idea of a \emph{constrained Delaunay triangulation} to curved surfaces. Evaluation on large datasets indicates that the method is both efficient and robust even for challenging models with low-quality triangulations. We explore how the curves and triangulations produced by the method facilitate a variety of tasks in digital geometry processing such as straightening cuts and segmentation boundaries, computing geodesic Bézier curves, and providing accurate boundary conditions for PDEs.