Discrete Curvature Flow for Hyperbolic 3-Manifolds with Complete Geodesic Boundaries
International Symposium on Visual Computing (ISVC2008)
Xiaotian Yin, Miao Jin, Feng Luo and Xianfeng Gu
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 3-
manifolds are valuable for the study of 3D topology and have the potential for
volumetric parameterization and shape matching.
This paper generalizes discrete curvature flow from surfaces to hyperbolic 3-
manifolds with complete geodesic boundaries. The metric deforms according to
the curvature, until the curvature is constant everywhere. The theoretical results
are introduced, the algorithm is explained in details, and thorough experiments are
carried out to demonstrate the effectiveness and efficiency of discrete 3-manifold
curvature flow.