Identifying the Critical Points of Skeleton-Based Convolution Surfaces for Conceptual Design

Abstract

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. 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 (3) and 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 surface. We describe an efficient algorithm that we have developed to find the critical points by analyzing the skeleton. The intent is to couple this algorithm with appropriate heuristics for determining parameter values of the convolution surface that will force its topology to match that of the skeleton.