Free-Form Modeling with Implicit Surface Patches

While it is possible to model a general closed surface of arbitrary genus as a single implicit surface patch, the geometry of such a global surface is di cult to specify, interactively control, and polygonize. The main di culties stem from the fact that implicit representations are iso-contours which generally have multiple real sheets, self{intersections and several other undesirable singularities. In this chapter I shall present details of several implicit polynomial surface splines (one is termed the A-patch) which overcome the above di culties, and show how these are used in C and C interpolation/approximation and interactive free-form modeling schemes. The potential of implicit patch splines remains largely latent and virtually all commercial and many research modeling systems are based on parametric spline representations. An exception is SHASTRA which allows modeling with both implicit and parametric splines [4]. The important issues in free-form patch modeling of shapes with arbitrary topology are: