GRAPHICAL MODELS AND IMAGE PROCESSING Vol. 58,No. 6,November,pp.544–552, 1996 ARTICLE NO . 0045 Rational Ruled Surfaces and Their Offsets H ELMUT POTTMANN AND WEI L U ¨ Institut fu ¨r Geometrie, Technische Universita ¨t Wien,Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria AND BAHRAM R AVANI Department of Mechanical and Aeronautical Engineering, University of California at Davis, Davis,California 95616 Received October 31, 1995;revised June 24, 1996;accepted July 16, 1996 ruled surfaces ofdegree 2m, whose shape is guided by m 1 1 control lines and m frame lines. This is an advantage In this paper, geometric design problems for rational ruled surfaces are studied. We investigate a line geometric control over the method of Ravani and Wang,which results in structure and its connection to the standard tensor product B- surfaces of degree 3m from the same data. spline representation, the use of the Klein model of line space, Recently,Sederberg and Saito [3] studied the Be ´zier and algorithms for geometry processing. The main part of the representation of planar sections of rational ruled surfaces paper is devoted to both classical and ‘‘circular’’ offsets of In our approach, the Klein model suggests a very simple rational ruled surfaces. These surfaces arise in NC milling. algorithm for the computation of the controlpoints of Excluding developable surfaces and, for circular offsets, certain planar section curves. Its geometric interpretation based conoidal ruled surfaces, we show that both offset types of ratio- on control null polarities also provides new geometric in- nal ruled surfaces are rational. In particular, we describe simple sight into planar sections and, dualto that,contours for tool paths which are rational quartics.  1996 Academic Press, Inc. central or parallel projection. More importantly, we also study offsets of rational ruled surfaces. In the past, Ravani and Ku [4] studied offsets of INTRODUCTION ruled surfaces. However,they only considered Bertrand A surface formed by a one-parameter set of straight offsets, which are not identical to constant distance offsets lines is a ruled surface. Ruled surfaces have been studied Their offset distance varies along a ruling as a function of extensively in classical geometry (see, for example, Edge the distance from the striction point of that ruling. Here, [1]).Although they have also been fundamental in early we study classical and circular offsets ofrationalruled works in CAGD (in terms of studies of lofted surfaces), surfaces. We prove that offsets of a rational (nondevelop- they have not been fully exploited for their applications able) ruled surface always admit a rational parameteriza- in geometric design and manufacturing. tion. This is an unexpected result and has practical implica A ruled surface can be designed as an interpolating sur- tions in applications where computations of offsets of ruled face of two boundary curves if one additionally prescribes surfaces will be important. a map between the curves and connects corresponding The organization of this paper is as follows. Section 1 points.Another design method uses tensor product sur- is an overview ofsome of the geometric fundamentals faces with one linear direction. Alternatively, a ruled sur- necessary for subsequent developments in the paper. Sec- face can be designed using a line geometric representation. tion 2 provides the details of our line geometric method The advantage of line geometric representation is that it for design of ruled surfaces. Section 3 provides the details leads to a curve type algorithm for design of ruled surfaces. of our results related to rational offsets of ruled surfaces. This was first shown by Ravani and Wang [2], who con- Finally, Section 4 describes patches of ruled surface offset structed ruled surfaces of degree 3m from m 1 1 control and low degree rational curves on them. lines. Here we also use a line geometric representation for 1. GEOMETRIC FUNDAMENTALS design of rational ruled surfaces. Our approach is based on the use of the Klein model of line space and is different Concepts from projective geometry and differential ge- ometry play a fundamental role in our paper. We assume from that of Ravani and Wang [2]. It results in rational 544 1077-3169/96 $18.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved.