Topological Consistency in Skeletal Modeling With Convolution Surfaces

A skeletal model consists of a collection of geometric primitives, such as points, lines, and triangles, that specifies the topology of an object. The skeleton is can be used to define surfaces, resulting in a solid model of the object. Of the various surface models that have been used for skeletal models, the convolution surface has some attractive properties. The convolution surface is an implicit surface that has been formulated for various geometric primitives such as lines, triangles, etc. Modeling with convolution surfaces guarantees a continuous and smoothly blended surface that is relatively easy to deform and intuitive to specify. However, when two skeleton elements are placed close to each other, unwanted blending can occur and the topology of the final shape becomes difficult to control. To be useful in conceptual engineering design, a skeletal modeler must ensure that the topology of surfaces generated agrees with that of the skeleton. Ensuring a match between the topology of the convolution surface and its skeleton is a challenging problem. In this paper, we propose a method based on Morse theory for ensuring topological consistency. By adjusting the parameters of the convolution surfaces, the topology of the final composite surface can be forced to match its root skeleton.