A study on surface and volume tiling for geometric modeling

Abstract

Surface tiling, as well as its counterpart in 3D, i.e. volume tiling, is a fundamental research problem in the subject of computer graphics and geometric modeling, which has found applications in numerous areas, such as computer-aided design (CAD), physical simulation, realtime rendering and architectural modeling. The objective of surface tiling is to compute discrete mesh representations for given surfaces which are often required to possess some desirable geometric properties. Likewise, volume tiling focuses on the study of discretizing a given 3D volume with complex boundary into a set of high-quality volumetric elements.

This thesis starts with the study of computing optimal sampling for parametric surfaces, that is, decompose the surface into quad patches such that 1) each quad patch should have their sides with equal length; and 2) the shapes and sizes of all the quad patches should be the same as much as possible. Then, the similar idea is applied to the discrete case, i.e. optimizing the face elements of a quad mesh surface with the goal of making it possess, as much as possible, face elements of desired shapes and sizes.

This thesis further studies the computation of hexagonal tiling on free-form surfaces, where the planarity of the faces is more concerned. Free-form meshes with planar hexagonal faces, to be called P-Hex meshes, provide a useful surface representation in discrete differential geometry and are demanded in architectural design for representing surfaces built with planar glass/metal panels. We study the geometry of P-Hex meshes and present an algorithm for computing a free-form P-Hex mesh of a specified shape.

Lastly, this thesis progresses to 3D volume case and proposes an automatic method for generating boundary-aligned all-hexahedron meshes with high quality, which possess nice numerical properties, such as a reduced number of elements and high approximation accuracy in physical simulation and mechanical engineering.