Constructing Regular Triangulation via Local Transformations: Theoretical and Practical Advances

Abstract

This thesis studies transformation between geometric structures via operations that act on some simplices of a geometric structure. Such a transformation is termed local transformation as an operation utilizes only information local to some neighboring simplices but nothing about attributes or configurations global in nature to the structure. Local transformation is simple to be implemented in practice and has been shown to be powerful and efficient to transform among various fundamental geometric structures. Such a transformation is also useful to repair geometric structures due to small adjustment to their simplices. For today's many-core architecture such as that of the GPU, local transformation is particularly attractive if it can be executed in parallel to gain good speedup at a low cost. Around local transformation, this thesis studies three local operations: flipping, splaying and twisting, and further develops a series of algorithms to compute regular triangulation and its dual structure convex hull.