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Computing general geometric structures on surfaces using Ricci flow

Authors:

Xianfeng David GuAuthors Info & Claims

Computer-Aided Design, Volume 39, Issue 8

Pages 663 - 675

https://doi.org/10.1016/j.cad.2007.05.006

Published: 01 August 2007 Publication History

Abstract

Systematically generalizing planar geometric algorithms to manifold domains is of fundamental importance in computer aided design field. This paper proposes a novel theoretic framework, geometric structure, to conquer this problem. In order to discover the intrinsic geometric structures of general surfaces, we developed a theoretic rigorous and practical efficient method, Discrete Variational Ricci flow. Different geometries study the invariants under the corresponding transformation groups. The same geometry can be defined on various manifolds, whereas the same manifold allows different geometries. Geometric structures allow different geometries to be defined on various manifolds, therefore algorithms based on the corresponding geometric invariants can be applied on the manifold domains directly. Surfaces have natural geometric structures, such as spherical structure, affine structure, projective structure, hyperbolic structure and conformal structure. Therefore planar algorithms based on these geometries can be defined on surfaces straightforwardly. Computing the general geometric structures on surfaces has been a long lasting open problem. We solve the problem by introducing a novel method based on discrete variational Ricci flow. We thoroughly explain both theoretical and practical aspects of the computational methodology for geometric structures based on Ricci flow, and demonstrate several important applications of geometric structures: generalizing Voronoi diagram algorithms to surfaces via Euclidean structure, cross global parametrization between high genus surfaces via hyperbolic structure, generalizing planar splines to manifolds via affine structure. The experimental results show that our method is rigorous and efficient and the framework of geometric structures is general and powerful.

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Cited By

Huang SWang HZhao YLin Z(2018)An analytical representation of conformal mapping for genus-zero implicit surfaces and its application to surface shape similarity assessmentComputer-Aided Design10.1016/j.cad.2015.02.00264:C(9-21)Online publication date: 30-Dec-2018
https://dl.acm.org/doi/10.1016/j.cad.2015.02.002
Xu GWang G(2008)Parametric polynomial minimal surfaces of degree six with isothermal parameterProceedings of the 5th international conference on Advances in geometric modeling and processing10.5555/1792279.1792305(329-343)Online publication date: 23-Apr-2008
https://dl.acm.org/doi/10.5555/1792279.1792305

Index Terms

Computing general geometric structures on surfaces using Ricci flow

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Information

Published In

cover image Computer-Aided Design

Computer-Aided Design Volume 39, Issue 8

August, 2007

89 pages

ISSN:0010-4485

Issue’s Table of Contents

Copyright © Elsevier Ltd © 2007.

Publisher

Butterworth-Heinemann

United States

Publication History

Published: 01 August 2007

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Cited By

Huang SWang HZhao YLin Z(2018)An analytical representation of conformal mapping for genus-zero implicit surfaces and its application to surface shape similarity assessmentComputer-Aided Design10.1016/j.cad.2015.02.00264:C(9-21)Online publication date: 30-Dec-2018
https://dl.acm.org/doi/10.1016/j.cad.2015.02.002
Xu GWang G(2008)Parametric polynomial minimal surfaces of degree six with isothermal parameterProceedings of the 5th international conference on Advances in geometric modeling and processing10.5555/1792279.1792305(329-343)Online publication date: 23-Apr-2008
https://dl.acm.org/doi/10.5555/1792279.1792305

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