Parwana, Sudheer Singh (2011)
Ph.D. thesis, University of Birmingham.
Surface triangulation is an active area of research due to its wide usage in a range of different computer aided applications, including computer aided design (CAD), manufacture (CAM) and finite element analysis (FEA). Although these applications are used to create, interrogate, manipulate and analyse surfaces, internally they actually approximate the surface geometry using a triangulation and then operate on the triangles, making them triangulation dependant algorithms. However, despite the reliance on the triangulation by the downstream application, there is very little work which has been focused on the inherent affects of the underlying triangulation on the performance or result of the application. Therefore, the impact of the triangulation on the downstream application is still not well known or defined. This thesis investigates triangulation and the downstream effects on the triangulation dependant method of surface flattening. Two novel topics are explored, notably right angle triangulation configurations (RATCs) and axis of minimum principal curvature (AMPC) influenced triangulations, each which was found to have an impact on the triangulation dependant method of surface flattening. Right angle triangles (RATs) are commonly used throughout surface flattening. However, given a set of uniformly sampled points, there are many different ways in which the diagonals can be placed to form a final triangulation consisting of RATs. These different configurations of edges are introduced as RATCs. To investigate the effects of RATCs on surface flattening, three global configurations are proposed; regular, diamond and chevron. In addition, local variations of RATCs are explored by fitting RATCs that best approximated the local Gaussian curvature of the original surface. Also considered is the influence of triangulation developability and its effects on flattening. Developable surfaces should flatten without inducing any area or shape distortion. However, it is shown that the transfer of this developabilty information from the surface to the triangulation can be lost. It is established that skewing the vertices of a triangulation so that no triangle edges followed along the AMPC, causes the flattening to induce distortion. As the shape of a triangulation is defined by its edges, if no edges follow along the AMPC, this means that the triangulation will not inherit this shape characteristic. Therefore, a new method is proposed to triangulate a surface, whilst ensuring that one edge of every triangle follows along the general direction of the AMPC. This thesis also introduces a new efficient and robust parametric trimmed surface triangulation method. Efficiency is gained during trimmed curve tracing by minimising the number of cells processed. Key features are the efficient tracing algorithm and knowledge of orientation of the trimming curves is not required. This approach also minimises the occurrence of degenerate triangles and copes with holes independently of the grid size.
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