Rational motion-based surface generation B. Ju ¨ttler a , M.G. Wagner b, * a Department of Mathematics, University of Technology, Schlobgartenstr. 7, 64289 Darmstadt, Germany b Department of Computer Science and Engineering, Arizona State University, Box 875406, Tempe, AZ 85287-5406, USA Received 17 April 1998; accepted 20 July 1998 Abstract This article is devoted to motion-based techniques for generating NURBS surfaces. We present a highly accurate approximation of the rotation-minimizing frame (RMF) of a space curve that leads to a RMF-based scheme for rational sweep surface modeling. Further we study envelopes of moving developable surfaces emphasizing the special cases of a moving cylinder or cone of revolution.  1999 Elsevier Science Ltd. All rights reserved. Keywords: Surface modeling; NURBS; Sweep surface; Enveloped surface; Rational motion 1. Introduction Motion-based surface generation is a fundamental princi- ple of shape construction. Surfaces can be generated by sweeping a profile curve (also called cross-section curve) along a given spine curve [8,15]. As a special case of this construction, one obtains the so-called pipe surfaces, which are generated by a moving circle. In contrast, surfaces can be generated as envelopes of moving objects. As an important special case one gets developable surfaces; they are the envelopes of moving planes [1]. Enveloped surfaces are particularly interesting in the context of milling and layered manufacturing, and also for the construction of gear tooth surfaces. The introduction of NURBS as the standard representa- tion for geometric data in CAD systems has made many new shape features available [9]. Using NURBS one may exactly describe surfaces of revolution, quadric surfaces such as ellipsoids and hyperboloids, developable surfaces, and also sweeping surfaces. Piecewise rational (or NURBS) motions [7] are the most appropriate tool for developing NURBS techniques for motion-based surface generation. The first part of the present article is devoted to sweeping surfaces. We present a rational approximation scheme for the rotation-minimizing frame (RMF) of a spine curve. In geometric modeling, this frame was introduced by Klok [8]. Based on biarc techniques, Wang and Joe [15] have recently developed an elegant approximation scheme. The newly developed rational scheme, as described later, improves the accuracy of the approximation to the RMF. It produces NURBS representations of sweeping surfaces that are pieced together of segments of degree (6, k), where k is the degree of the profile (or cross-section) curve. Unlike the biarc scheme, it produces a true C 1 motion, which gives smooth sweeping surfaces also for non-planar profile curves. The scheme can readily be modified to generate sweeping surfaces matching more general input data, such as a sequence of positions of the profile curve. The second part of the article deals with the construc- tion of envelopes of moving developable surfaces. Developable surfaces are envelopes of one-parameter sets of planes. This includes cylinders and cones as degenerate cases. If both the moving surface and the motion are rational, the resulting envelope will be a rational TP NURBS surface. This extends the ideas introduced in Ref. [7] for moving polyhedra to a much more general surface type. 2. Basics from kinematics The points of Euclidean 3-space are described by their coordinates p  p 1 p 2 p 3  T with respect to a Cartesian coordinate system. Sometimes, however, it will be advanta- geous to use the homogeneous coordinates p  (p 0 p 1 p 2 p 3 ) T instead of, p  (0 0 0 0) T . If the homogeneous coordinates of a point are given, then the corresponding Cartesian coordinates are p i  p i =p 0 ; i  1,2,3. Conversely, the possible homogeneous coordinates of the point p are p ll p 1 l p 2 l p 3  T with l  R, l  0. The coefficient Computer-Aided Design 31 (1999) 203–213 COMPUTER-AIDED DESIGN 0010-4485/99/$ - see front matter  1999 Elsevier Science Ltd. All rights reserved. PII: S0010-4485(99)00016-0 * Corresponding author. Tel.: +1-602-965-1735; fax: +1-602-965- 5142. E-mail address: wagner@asu.edu (M.G. Wagner)