IMPLICIT AND PARAMETRIC CURVES AND SURFACES FOR COMPUTER AIDED GEOMETRIC DESIGN

The following two problems are shown to have closed form solutions requiring only the arithmetic operations of addition, subtraction, multiplication and division. First, given a curve or surface defined parametrically in terms of rational polynomials, find an implicit polynomial equation which defines the same curve or surface. Second, given the cartesian coordinates of a point on such a curve or surface, find the parameter(s) corresponding to that point. The first problem has not seen a solution in the computer aided geometric design literature, and the second problem has not seen a closed form solution. The significance of these results is demonstrated by their application to the current problems facing the field of computer aided geometric design of computing intersections of parametric curves and surfaces. Several applications of these results are made to various intersection problems. A free form surface patch suitable for many computer aided geometric design applications is introduced. This patch is called the Steiner surface patch, and can be defined as a rational quadratic Bezier triangle. It is shown to be a degree four algebraic surface, that it can degenerate into any quadric surface, and that any plane will intersect it in a parametric curve.