Computational techniques for processing parametric surfaces

TL;DR: This paper presents computational techniques using which subdivision algorithms may be devised for the processing (rendering, intersection detection, silhouette detection) of parametrically defined surfaces by subd dividing surface patches until they are simple enough for direct handling.

Abstract: This paper presents computational techniques using which subdivision algorithms may be devised for the processing (rendering, intersection detection, silhouette detection) of parametrically defined surfaces. These algorithms work by subdividing surface patches until they are simple enough for direct handling. For example, planar surface patches can be handled analytically. For interference it is necessary to box the surface as well. The computational techniques presented are essentially for efficient computation of surface properties needed by the processing tasks. The three properties considered are: (1) Euclidean bounds: this is done by working in extrema in x, y, and z over the patch; (2) planarity estimate: this test is defined in terms of the linearity of constituent curves; (3) Local visibility: which says whether a patch is totally visible, invisible, or partially visible from a given viewpoint. Rendering algorithms make use of this information. This too is done by working in extrema of the visibility function. All the techniques are based on the parametric form of the surface representation. The class of surfaces that can be handled by these techniques is very large, basically C2 continuous surfaces. A class of surfaces known as product surfaces is specially introduced as the above methods are extremely efficient for this class. Application of these methods to bicubic surfaces is also discussed.