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An algorithm for the construction of intrinsic delaunay triangulations with applications to digital geometry processing

Published: 30 July 2006 Publication History

Abstract

The discrete Laplace-Beltrami operator plays a prominent role in many Digital Geometry Processing applications ranging from denoising to parameterization, editing, and physical simulation. The standard discretization uses the cotangents of the angles in the immersed mesh which leads to a variety of numerical problems. We advocate use of the intrinsic Laplace-Beltrami operator. It satis- fies a local maximum principle, guaranteeing, e.g., that no flipped triangles can occur in parameterizations. It also leads to better conditioned linear systems. The intrinsic Laplace-Beltrami operator is based on an intrinsic Delaunay triangulation of the surface. We give an incremental algorithm to construct such triangulations together with an overlay structure which captures the relationship between the extrinsic and intrinsic triangulations. Using a variety of example meshes we demonstrate the numerical benefits of the intrinsic Laplace-Beltrami operator.

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cover image ACM Conferences
SIGGRAPH '06: ACM SIGGRAPH 2006 Courses
July 2006
83 pages
ISBN:1595933646
DOI:10.1145/1185657
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 30 July 2006

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  • (2024)Differentiable Geodesic Distance for Intrinsic Minimization on Triangle MeshesACM Transactions on Graphics10.1145/365812243:4(1-14)Online publication date: 19-Jul-2024
  • (2024)A Comprehensive Survey on Delaunay Triangulation: Applications, Algorithms, and Implementations Over CPUs, GPUs, and FPGAsIEEE Access10.1109/ACCESS.2024.335470912(12562-12585)Online publication date: 2024
  • (2023)Surface Simplification using Intrinsic Error MetricsACM Transactions on Graphics10.1145/359240342:4(1-17)Online publication date: 26-Jul-2023
  • (2023)ARAP Revisited Discretizing the Elastic Energy using Intrinsic Voronoi CellsComputer Graphics Forum10.1111/cgf.1479042:6Online publication date: 4-Apr-2023
  • (2021)Integer coordinates for intrinsic geometry processingACM Transactions on Graphics10.1145/3478513.348052240:6(1-13)Online publication date: 10-Dec-2021
  • (2020)Dirichlet energy of Delaunay meshes and intrinsic Delaunay triangulationsComputer-Aided Design10.1016/j.cad.2020.102851126(102851)Online publication date: Sep-2020
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