Detecting and quantifying envelope singularities in the plane

The mathematical envelopes of families of both rigid and non-rigid moving shapes play a fundamental role in a variety of problems from very diverse application domains, from engineering design and manufacturing to computer graphics and computer assisted surgery. Geometric singularities in these envelopes are known to induce malfunctions or unintended system behavior, and the corresponding theoretical and computational difficulties induced by these singularities are not only massive, but also well documented. We describe a new approach to detect and quantify the envelope singularities induced by 2-dimensional shapes of arbitrary complexity moving according to general non-periodic and non-singular planar affine motions. Our approach, which does not require any envelope computations, is reframing the problem in terms of “fold points” and “fold regions” in the neighborhood of geometric singularities, and we show that the existence of these fold points is a necessary condition for the existence of singularities. We establish a mathematically well defined duality between the 2-dimensional Euclidean space in which the motion takes place and a 2+1 spacetime domain. Based on this duality, we recast the problem of detecting and quantifying geometric singularities into inherently parallel tests against the original geometric representation in the 2-dimensional Euclidean space. We conclude by discussing the significance of our results, and the extension of our approach to 3-dimensional moving shapes.