Determine When a Parametric Surface is a Surface of Revolution

Abstract

A surface of revolution is a surface that can be generated by rotating a planar curve (the directrix) around a straight line (the axis) in the same plane. Using the mathematics of quaternions, we provide a parametric equation of a surface of revolution generated by rotating a directrix about an axis by quaternion multiplication of the parametric representations of the directrix curve and the line of axis. Then, we describe an algorithm to determine whether a parametric surface is a surface of revolution, and identify the axis and the directrix. Examples are provided to illustrate our algorithm.