Local Metrics for Rigid Body Displacements

Abstract

One hundred years ago, Eduard Study introduced a very elegant method to describe a rigid body displacement in three-space. He mapped each position of a rigid body onto a point on a quadric, now called the Study quadric. This quadric is a six-dimensional rational hyper-surface, embedded in a seven-dimensional projective real space, called Study’s soma space. More than half a century later Ravani and Roth reconfigured Study’s soma space into a three-dimensional dual projective space, and defined a geometric metric for rigid body displacements. Here, approximately 20 years later, we again use Study’s quadric and define a new metric for rigid body displacements based on an optimized local mapping of the quadric. The local mappings of the quadric are achieved using stereographic projections, resulting in an affine space where the Euclidean definition of a metric can be used for rigid body displacements and techniques from design of curves and surfaces can be directly utilized for motion design. The results are illustrated by examples.