Discrete Curvature Flows for Surfaces and 3-Manifolds

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:
Emerging Trends in Visual Computing
Year:
2009

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Abstract

Intrinsic curvature flows can be used to design Riemannian metrics by prescribed curvatures. This chapter presents three discrete curvature flow methods that are recently introduced into the engineering fields: the discrete Ricci flow and discrete Yamabe flow for surfaces with various topology, and the discrete curvature flow for hyperbolic 3-manifolds with boundaries. For each flow, we introduce its theories in both the smooth setting and the discrete setting, plus the numerical algorithms to compute it. We also provide a brief survey on their history and their link to some of the engineering applications in computer graphics, computer vision, medical imaging, computer aided design and others.