Self-intersections of offsets of quadratic surfaces: Part I, explicit surfaces

Abstract

Although offset surfaces are widely used in various engineering applications, their degenerating mechanism is not well known in a quantitative manner. Offset surfaces are functionally more complex than their progenitor surfaces and may degenerate even if the progenitor surfaces are regular. Self-intersections of the offsets of regular surfaces may be induced by concave regions of surface where the positive offset distance exceeds the maximum absolute value of the negative minimum principal curvature or the absolute value of the negative offset distance exceeds the maximum value of the positive maximum principal curvature. It is well known that any regular surface can be locally approximated in the neighborhood of a pointp by the explicit quadratic surface of the form r(x,y)=[x,y1/2(αx²+βy²)]^T to the second order where −α and −β are the principal curvatures at pointp. Therefore investigations of the selfintersecting mechanisms of the offsets of explicit quadratic surfaces due to differential geometry properties lead to an understanding of the self-intersecting mechanisms of offsets of regular parametric surfaces. In this paper, we develop the equations of the self-intersection curves of an offset of an explicit quadratic surface. We also develop an algorithm to detect and trace a small loop of a self-intersection curve of an offset of a regular parametric surface based on our analysis of offsets of explicit quadratic surfaces. Examples illustrate our method.