Implicit Surfaces for Geometric Modeling

The rst section will build some basics. It will contrast implicit and parametric methods, and overview the broad range of implicit surfaces used today. It will develop fundamental properties of quadratic implicit polynomial surfaces, and illustrate how these surfaces have been traditionally used in CAD systems for handling \prismatic" objects (e.g. piston rods), and how recent research on algebraic patches has extended their power to now support \free-form" objects (e.g. bones). This is followed by an introduction to the class of implicit skeletal methods that build shapes using distance functions and blends. As a new twist, this section will develop formal notions of representation schemes (particularly, Brep and CSG), conversions (fundamental problems of separation and describability), and their impact on rendering algorithms. This leads to an exercise in designing direct rendering hardware for quadratic surfaces, both for Brep-based and CSG-based parallel processing. In particular, the architecture of the prototype RayCasting Engine (RCE), with over 2K parallel processors, for direct CSG raycasting will be discussed.