Use of Free-Form Low-Degree Algebraic Surfaces in Solid Modeling

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The fields of surface modeling and solid modeling, which are likely to form the basis for future computer-aided design (CAD) systems, have been developing independently. Today users of systems based on one or the other technique find they require the functionality of both types of systems simultaneously. The goal of this research is to provide a technique for modeling free-form surfaces in a manner which facilitates their incorporation in solid modeling systems. The surfaces used in this technique will be described by algebraic functions defined within a parallelepiped. By using control points contained in the parallelepiped, the algebraic function can be expressed with B-splines, automatically allowing for continuity throughout the surface. These algebraic surfaces can be implemented easily in solid modelers since the implicit surface equation defines specific half- spaces. A useful free-form primitive will be obtained by clipping them against the parallelepiped and then rendering as solids. Ray tracing methods will be efficiently applied to these surfaces, since the ray intersection equation is a univariate polynomial whose degree is the degree of the surface.