Topological consistency in skelatal modeling with convolution surfaces for conceptual design
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This dissertation describes a new topology analysis tool for a skeletal based geometric modeling system for conceptual design. Skeletal modeling is an approach to creating solid models in which the engineer designs with lower dimensional primitives such as points, lines, and triangles. The skeleton is then "skinned over" to create the surfaces of the three-dimensional object. In this research, convolution surfaces are used to provide the flesh to the skeleton. Convolution surfaces are generated by convolving a kernel function with a geometric field function to create an implicit surface. Certain properties of convolution surfaces make them attractive for skeletal modeling, including: (1) providing analytic solutions for various geometry primitives (including points, line segments, and triangles); (2) generating smooth surfaces; and (3) providing well-behaved blending. We assume that engineering designers expect the topology of a skeletal model to be identical to that of the underlying skeleton. However, the topology of convolution surfaces can change arbitrarily, making it difficult to predict the topology of the generated surface from knowledge of the topology of the skeleton. To address this issue, we apply Morse theory to analyze the topology of convolution surfaces by detecting the critical points of the surfaces. We developed an efficient and intelligent algorithm to find the critical points (CPs) by analyzing the skeleton. The critical points provide valuable information about the topology of the convolution surfaces. By tracking the CPs, we know where and what kind of topology changes happen when the threshold value reaches the critical value at the CP. Topology matching is done in two steps: (1) global topology is tested by comparing the Betti numbers (number of component, loops, and voids) of the skeleton and the generated convolution surfaces; (2) with matched Betti numbers, local topology is tested by comparing the location of each loop and void area between the skeleton and surfaces. If the topology does not match, appropriate heuristics for determining parameter values of the convolution surfaces are applied to force the surface topology to match that of the skeleton. A recommend threshold value is then provided to generate the topology matched convolution surfaces.