DefinePK

Abstract

Constructing a new surface from a given regular surface is an important task in Surface Modeling. A parallel surface (called also an offset surface) of a polynomial or rational surface is widely used in computer-aided design. The shape of a regular surface can be characterized by its principle curvatures, or equivalently, by its Gaussian and mean curvatures. The parabolic cylinder is a regular polynomial surface with many applications in product design. Parabolic-trough solar collectors and parabolic cylindrical reflector antennas are good illustrations of the integrated form and function. In this paper, we present constructions of two cylindrical surfaces associated with an arbitrary parabolic cylinder. The first one is the unique focal surface of the parabolic cylinder which is defined by the reciprocal of the unique nonzero principle curvature. Its set of singular points coincides with the points of a straight line. The second one is a new regular surface obtained by the normal and reciprocal mean curvature of the original parabolic cylinder. This surface, called a generalized focal surface, consists of parabolic and planar points. A comparison of the properties of the parabolic cylinder, its focal surface, and its generalized focal surface is given. The relative positions of these three surfaces are also discussed.