ELSEVIER Computer Aided Geometric Design 13 (1996) 621-628 COMPUTER AIDED GEOMETRIC DESIGN Pipe surfaces with rational spine curve are rational Wei LO 1, Helmut Pottmann *, lnstitut f&'r Geometrie, Technische Universitfit Wien, Wiedner Hauptstrasse 8-10, A-1040 Wien, Austria Received September 1995 Abstract The envelope of the one-parameter set of congruent spheres which are centered at the points of a curve S is a pipe surface with ~ as spine curve. We prove that pipe surfaces with rational spine curve always admit a rational parameterization and propose an algorithm for its computation. Keywords: Rational surface; Rational parameterization; Pipe surface; Offset surface; Normal ringed surface; Blend surface; Pythagorean-hodograph curve I. Introduction It is well-known that the offsets of a planar rational curve are in general not rational and therefore need to be approximated by NURBS curves for compatibility with current CAD/CAM systems (Farin, 1994). This motivated a series of contributions on rational curves with rational offsets, which are also called rational Pythagorean-hodograph (PH) curves in view of the method used by Farouki and Sakkalis (1990) in order to construct polynomial curves with rational offsets (Ait Haddou and Biard, 1994; Albrecht and Farouki, 1996; Farouki, 1992, 1994; Farouki and Neff, 1995; Fiorot and Gensane, 1994; L0, 1994, 1995; Pottmann, 1994, 1995a, 1995b). Rational space curves S = 8(t) = (z(t), y(t), z(t)) with rational parametric speed, i.e., z '2 (t) + y,2 (t) + z '2 (t) = w 2 (t) for some rational function w(t), have been studied by Farouki and Sakkalis (1994) and Pottmann (1994). It has been shown that the envelope of the one-parameter set of congruent spheres, centered at the points of such a PH space curve S, is a rational pipe surface. Pipe surfaces may be considered as spatial analogue to offset curves in the plane. However, the PH property of the spine curve S is not * Corresponding author. E-mail: pottmann@geometrie.tuwien.ac.at. 1Permanent address: Department of Applied Mathematics, Zhejiang University, Hangzhou 310027, P.R. China. 0167-8396/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0167-8396(95)00051-8