Ruled Surfaces of Revolution with Moving Axes and Angles

Abstract

A ruled surface of revolution with moving axes and angles is a rational tensor product surface generated from a line and a rational space curve by rotating the line (the directrix) around vectors and angles generated by the rational space curve (the director). Only right circular cylinders and right circular cones are ruled surfaces that are also surfaces of revolution, but we show that a rich collection of other ruled surfaces such as hyperboloids of one sheet, 2-fold Whitney umbrellas, and a wide variety other interesting ruled shapes are ruled surfaces of revolution with moving axes and angles. We present a fast way to compute the implicit equation of a ruled surface of revolution with moving axes and angles from two linearly independent vectors that are perpendicular to the directrix of the surface. We also provide an algorithm for determining whether or not a given rational ruled surface is a ruled surface of revolution with moving axes and angles.