Discrete Metric Design and Discrete Tangent Bundles: from Surfaces to 3-Manifolds

Abstract

One of the fundamental tasks in geometric modeling and computer graphics is to study shapes, such as surfaces and volumes, and differential objects associated with them, such as vector fields. For the study of shapes, the challenge in many cases comes from the need of a parameter domain that has a canonical and simple shape, which is equivalent to designing a metric of special properties. For differential objects, a fundamental problem is how to represent tangent bundles in a discrete setting, so that covariant differentiation and connections can be computed accordingly. This dissertation aims to design rigorous and practical methods to deal with both tasks. For discrete metric design, a global parameterization method, called slit map, is designed for genus zero surfaces with multiple holes using discrete differential forms. A second method is designed to map volumetric handle bodies to direct product domains. A third method is designed to compute constant curvature metrics for hyperbolic 3-manifolds using a discrete curvature flow. For discrete tangent bundles, we propose discrete constructions using tetrahedral meshes to represent unit tangent bundles for various surfaces, including topological disks and closed orientable surfaces of arbitrary genus. All the proposed methods are based on solid results from topology and differential geometry, and are adapted with best efforts to engineering problems ranging from surfaces to 3-manifolds.