Working with ruled surfaces in solid modeling

Abstract

The interplay between algebraic geometry and graphics/solid modeling is a natural and strong one. This paper addresses the topic of ruled surfaces, a class that has long been of interest to the mathematical community, and brings it more squarely into the realm of computer science by giving constructive algorithms for ruled surfaces. These algorithms allow ruled surfaces to be used more easily in a solid modeling system. Specifically, we show (a) how to identify that a surface is ruled from its equation (b) how to find the generator through a given point of a ruled surface and (c) how to find a directrix curve of a ruled surface. As an example of how thelle algorithms can be put to use in a solid modeling environment, we show how to parameterize a ruled surface. Ruled surfaces share properties of both curves and Burfaces, which make ruled surfaces a very useful class in the difficult transition between curves and surfaces in solid modeling. They can be used to extend algorithms for curves (which are easier to develop) to algorithms for surfaces. The mathematical theory of curves and surfaces can continue to guide their incorporation into solid modelers although, as is shown in this paper, computer scientists will often have to develop constructive techniques to replace existential mathematical statements.