Interpolation by polygon rolling motions for approximate sweep computation

Abstract

We interpolate planar rigid body motions by piecewise rotational and translational motions. The trajectories of all points are then arc splines, i.e., curves composed of circular arcs or line segments. For objects with arc spline boundaries, the boundary of the volume swept by the object and the motion is then again an arc spline. We prove that the distance between the original motion and its approximation as between curves in a suitable kinematic image space converges quadratically to zero. This implies the same rate of convergence between point trajectories under the two motions.