Discrete Curvature Flow for Surfaces and 3-Manifolds
نویسندگان
چکیده
This chapter introduces the concepts, theories and algorithms of discrete curvature flows for surfaces with arbitrary topologies. Discrete curvature flow for hyperbolic 3-manifolds with geodesic boundaries is also presented. Curvature flow method can be used to design Riemannian metrics by prescribed curvatures, and applied for parameterization in graphics, shape registration in computer vision, brain mapping in medical imaging, spline construction in computer aided geometric design, and many other engineering fields.
منابع مشابه
Discrete Curvature Flow for Hyperbolic 3-Manifolds with Complete Geodesic Boundaries
Every surface in the three dimensional Euclidean space have a canonical Riemannian metric, which induces constant Gaussian curvature and is conformal to the original metric. Discrete curvature flow is a feasible way to compute such canonical metrics. Similarly, three dimensional manifolds also admit canonical metrics, which induce constant sectional curvature. Canonical metrics on 3manifolds ar...
متن کاملDiscrete Curvature Flows for Surfaces and 3-Manifolds
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 3manifolds with boundaries. For each flow, we introduc...
متن کاملGEOMETRIZATION OF HEAT FLOW ON VOLUMETRICALLY ISOTHERMAL MANIFOLDS VIA THE RICCI FLOW
The present article serves the purpose of pursuing Geometrization of heat flow on volumetrically isothermal manifold by means of RF approach. In this article, we have analyzed the evolution of heat equation in a 3-dimensional smooth isothermal manifold bearing characteristics of Riemannian manifold and fundamental properties of thermodynamic systems. By making use of the notions of various curva...
متن کاملOn actions of discrete groups on nonpositively curved spaces
This note contains several observations concerning discrete groups of non-parabolic isometries of spaces of nonpositive curvature. We prove that (almost all) surface mapping class groups do not admit such actions; in particular, they do not act cocompactly. A new obstruction to the existence of a nonpositively curved metric on closed manifolds is presented. We give examples of 4-manifolds bered...
متن کاملA Combinatorial Curvature Flow for Compact 3-manifolds with Boundary
We introduce a combinatorial curvature flow for piecewise constant curvature metrics on compact triangulated 3-manifolds with boundary consisting of surfaces of negative Euler characteristic. The flow tends to find the complete hyperbolic metric with totally geodesic boundary on a manifold. Some of the basic properties of the combinatorial flow are established. The most important one is that th...
متن کامل