Keywords

computational geometry, computer graphics, interpolation, polynomials, base points, cubic algebraic surfaces, free-form modeling, implicit polynomial equation, intersection locus, polygonization algorithm, power basis representation, space curve

Abstract

The tutorial presents some tools for free-form modeling with algebraic surfaces, that is, surfaces that can be defined using an implicit polynomial equation f(x, y, z )=0. Cubic algebraic surfaces (defined by an implicit equation of degree 3) are emphasized. While much of this material applies only to cubic surfaces, some applies to algebraic surfaces of any degree. This area of the tutorial introduces terminology, presents different methods for defining and modeling with cubic surfaces, and examines the power basis representation of algebraic surfaces. Methods of forcing an algebraic surface to interpolate a set of points or a space curve are also discussed. The parametric definition of cubic surfaces by imposing base points is treated, along with the classical result that a cubic surface can be defined as the intersection locus of three two-parameter families of planes. Computer-generated images of algebraic surfaces created using a polygonization algorithm and Movie. BYU software illustrate the concepts presented.

Original Publication Citation

Sederberg, T. W. "Techniques for Cubic Algebraic Surfaces I." Computer Graphics and Applications, IEEE 1.4 (199): 14-25.