Kinematic Convexity of Rigid Body Displacements

In this paper, we explore the notion of kinematic convexity for rigid body displacements. Previously, we have shown that when spatial rigid body displacements are represented by dual quaternions, an oriented projective space is better suited for the image space of displacements. Geometric algorithms for rigid body motions become more general and elegant when developed from the perspective of oriented projective geometry. By extending the concept of convexity in affine geometry to oriented projective geometry of the image space of rigid body displacements, we define the concept of kinematic convexity. This concept, apart from being theoretically significant, facilitates localization of a displacements and provides a measure of the kinematic separation useful in collision prediction, interference checking, and geometric analysis of swept volumes.