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A Smoothness Energy without Boundary Distortion for Curved Surfaces

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Eitan GrinspunAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 39, Issue 3

Article No.: 18, Pages 1 - 17

https://doi.org/10.1145/3377406

Published: 18 March 2020 Publication History

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Abstract

Current quadratic smoothness energies for curved surfaces either exhibit distortions near the boundary due to zero Neumann boundary conditions or they do not correctly account for intrinsic curvature, which leads to unnatural-looking behavior away from the boundary. This leads to an unfortunate trade-off: One can either have natural behavior in the interior or a distortion-free result at the boundary, but not both. We introduce a generalized Hessian energy for curved surfaces, expressed in terms of the covariant one-form Dirichlet energy, the Gaussian curvature, and the exterior derivative. Energy minimizers solve the Laplace-Beltrami biharmonic equation, correctly accounting for intrinsic curvature, leading to natural-looking isolines. On the boundary, minimizers are as-linear-as-possible, which reduces the distortion of isolines at the boundary. We discretize the covariant one-form Dirichlet energy using Crouzeix-Raviart finite elements, arriving at a discrete formulation of the Hessian energy for applications on curved surfaces. We observe convergence of the discretization in our experiments.

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Index Terms

A Smoothness Energy without Boundary Distortion for Curved Surfaces
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
      1. Mesh geometry models
2. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
      1. Partial differential equations
    2. Numerical analysis
      1. Discretization
      2. Numerical differentiation

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cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 39, Issue 3

June 2020

179 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/3388953

Editor:
Marc Alexa
TU Berlin, Germany

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Publication History

Published: 18 March 2020

Accepted: 01 January 2020

Revised: 01 January 2020

Received: 01 May 2019

Published in TOG Volume 39, Issue 3

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