The geometry of the point-paths generated by rigid-body motion in two and three dimensions

Abstract

The geometry of the point-paths of planar and spatial rigid-body motions are studied using techniques of differential geometry. In particular, we look at sets of points which, at an instant, display some particular behaviour. In the planar case this involves looking at sets of points of zero curvature (inflection points) and of stationary curvature (vertices). For spatial motion we look at sets of points of zero curvature and of zero torsion. We then examine how these sets degenerate when the motion itself has special behaviour. For the spatial motions we detail a method of putting the motion into a normal form which enables calculations to be made more easily and classify the motions according to the type of normal form that results. We then use these classifications to describe the degeneracies of the zero-curvature curve and the zero-torsion surface. We also give a brief description of some of the other methods which can be used to represent spatial motion.