Computing Rational Parametrizations of Canal Surfaces

Current CAD systems can represent curves and surfaces only in rational B-spline (NURBS) form ( .Farin, 1994; .Hoschek and Lasser, 1993). On the other hand, certain curves and surfaces that arise in practical applications such as offsets of rational curves or surfaces are in general not rational and therefore need to be approximated. This motivated .Farouki and Sakkalis (1990) to introduce the so-called Pythagorean-hodograph (PH) curves, which are planar polynomial curves that possess rational offsets. Recent research on PH curves and their generalizations to the full class of rational curves with rational offsets has shown that they are well–suited for practical use (see e.g. .Ait Haddou and Biard, 1995; .Albrecht and Farouki, 1995; .Farouki, 1992; .Lü, 1995a; .Pottmann, 1995a, .b and the references therein). The offset at distance r to a curve m(t) in 3-space can be defined as the envelope of the set of spheres with radius r which are centered at m(t). Such a surface is called a pipe surface or tubular surface with spine curve m(t). Surprisingly, it turned out that pipe surfaces with rational spine curve m(t) always admit a rational parameterization ( .Lü and Pottmann, 1996). In the present paper, we will generalize this result as follows. Canal surfaces, defined as envelope of a one-parameter set of spheres with a rational radius function r(t) and centers at a rational curve m(t) can be rationally parametrized. A constructive proof for this result is given, along with other techniques to compute rational parametrizations of pipe and canal surfaces with low degree spine curves and radius functions. In practical applications, canal surfaces mainly appear as blend surfaces and transition surfaces between pipes. Note that the present class of surfaces contains as special case the Dupin cyclides, which have been proposed by several authors for various applications in Computer Aided Geometric Design (see e.g. .Pratt, 1995; .Srinivas and Dutta, 1994).