Computing and Visualizing Constant-Curvature Metrics on Hyperbolic 3-Manifolds with Boundaries

Authors:
Xiaotian Yin;Miao Jin;Feng Luo;Xianfeng David Gu
Affiliations:
Computer Science Department, State University of New York at Stony Brook,;Center for Advanced Computer Studies, University of Louisiana at Lafayette,;Department of Mathematics, Rutgers University,;Computer Science Department, State University of New York at Stony Brook,
Venue:
ISVC '08 Proceedings of the 4th International Symposium on Advances in Visual Computing
Year:
2008

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Abstract

Almost all three dimensional manifolds admit canonical metricswith constant sectional curvature. In this paper we proposed a newalgorithm pipeline to compute such canonical metrics for hyperbolic3-manifolds with high genus boundary surfaces. The computation isbased on the discrete curvature flow for 3-manifolds, where themetric is deformed in an angle-preserving fashion until thecurvature becomes uniform inside the volume and vanishes on theboundary. We also proposed algorithms to visualize the canonicalmetric by realizing the volume in the hyperbolic spaceℍ3, both in single period and in multiple periods.The proposed methods could not only facilitate the theoreticalstudy of 3-manifold topology and geometry using computers, but alsohave great potentials in volumetric parameterizations, 3D shapecomparison, volumetric biomedical image analysis and etc.