A Dual Ellipse Is a Cylindroid

Abstract

Abstract In this paper, we take a relook at two-degree-of-freedom instantaneous rigid body kinematics in terms of dual numbers and vectors, and show that a dual ellipse is a cylindroid. The instantaneous angular and linear velocities of a rigid body is expressed as a dual velocity vector, and the inner product of two dual vectors, as a dual number, is used. We show that the tip of a dual velocity vector lies on a dual ellipse, and the maximum and minimum magnitude of the dual velocity vector, for a unit speed motion, can be obtained as eigenvalues of a positive definite, symmetric matrix whose elements are the dual numbers from the inner products. From the real and dual parts of the equation of the dual ellipse, we derive the equation of a cylindroid (Ball,1900).