Geometric computations with algebraic varieties of bounded degree

Abstract

The set of solutions to a collection of polynomial equations is referred to as an algebraic set. An algebraic set that cannot be represented as the union of two other distinct algebraic sets, neither containing the other, is said to be irreducible. An irreducible algebraic set is also known as an algebraic variety. This paper deals with geometric computations with algebraic varieties. The main results are algorithms to (1) compute the degree of an algebraic variety, (2) compute the rational parametric equations (a rational map from points on a hyperplane) for implicitly defined algebraic varieties of degrees two and three. These results are based on sub-algorithms using multi-polynomial resultants and multi-polynomial remainder sequences for constructing a one-to-one projection map of an algebraic variety to a hypersurface of equal dimension, as well as, an inverse rational map from the hypersurface to the algebraic variety. These geometric computations arise naturally in geometric modeling, computer aided design, computer graphics, and motion planning, and have been used in the past for special cases of algebraic varieties, i.e. algebraic curves and surfaces.