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Quad-mesh based isometric mappings and developable surfaces

Published: 12 August 2020 Publication History

Abstract

We discretize isometric mappings between surfaces as correspondences between checkerboard patterns derived from quad meshes. This method captures the degrees of freedom inherent in smooth isometries and enables a natural definition of discrete developable surfaces. This definition, which is remarkably simple, leads to a class of discrete developables which is much more flexible in applications than previous concepts of discrete developables. In this paper, we employ optimization to efficiently compute isometric mappings, conformal mappings and isometric bending of surfaces. We perform geometric modeling of developables, including cutting, gluing and folding. The discrete mappings presented here have applications in both theory and practice: We propose a theory of curvatures derived from a discrete Gauss map as well as a construction of watertight CAD models consisting of developable spline surfaces.

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cover image ACM Transactions on Graphics
ACM Transactions on Graphics  Volume 39, Issue 4
August 2020
1732 pages
ISSN:0730-0301
EISSN:1557-7368
DOI:10.1145/3386569
Issue’s Table of Contents
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 the author(s) 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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Publication History

Published: 12 August 2020
Published in TOG Volume 39, Issue 4

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Author Tags

  1. computational fabrication
  2. computer-aided design
  3. developable spline surface
  4. developable surface
  5. discrete differential geometry
  6. discrete isometry
  7. shape optimization

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  • (2024)All you need is rotation: Construction of developable stripsACM Transactions on Graphics10.1145/368794743:6(1-15)Online publication date: 19-Dec-2024
  • (2024)Quad mesh mechanismsACM Transactions on Graphics10.1145/368793943:6(1-17)Online publication date: 19-Dec-2024
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  • (2024)Design of optimal quasi-developable surface via simulated annealing based shape-parameter-search algorithmAnnals of Mathematics and Artificial Intelligence10.1007/s10472-024-09962-6Online publication date: 19-Dec-2024
  • (2023)Developable Quad Meshes and Contact Element NetsACM Transactions on Graphics10.1145/361835542:6(1-13)Online publication date: 5-Dec-2023
  • (2023)Evolutionary Piecewise Developable ApproximationsACM Transactions on Graphics10.1145/359214042:4(1-14)Online publication date: 26-Jul-2023
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