M. E. Hohmeyer
Abstract
The parametric, geometric, or Frenet frame continuity of a rational curve has often been ensured by requiring the homogeneous polynomial curve associated with the rational curve to possess either parametric, geometric, or Frenet frame continuity, respectively. In this paper, we show that this approach is overly restrictive and derive the constraints on the associated homogeneous curve that are both necessary and sufficient to ensure that the rational curve is either parametrically, geometrically, or Frenet frame continuous.
- 1 BARSKY, B.A. The Beta-spline: A local representation based on shape parameters and fundamental geometric measures. Ph.D. thesis, Dept. of Computer Science, Univ. of Utah, Salt Lake City, Utah, Dec. 1981. Google Scholar
- 2 BARSK~, B.A. Introducing the rational Beta-spline. In Proceedings of the 3rd International Conference on Engineering Graphics and Descriptive Geometry, Vol. 1 (Vienna, Austria, July 11- 16, 1988). American Society for Engineering Education, 1988, pp. 16-27.Google Scholar
- 3 BARSKV, B.A. Computer Graphics and Geometric Modeling Using Beta-splines. Springer-Verlag, Heidelberg, 1988. Google Scholar
- 4 BAr~SKY, B.A. The rational Beta-spline form for curve and surface representation. Submitted for publication.Google Scholar
- 5 BARSKY, B. A., AND DERO~ClE, T.D. Geometric continuity of parametric curves. Tech. Pep. UCB/CSD 84/205, Computer Science Division, Electrical Engineering and Computer Sciences Dept., Univ. of California, Berkeley, Oct. 1984. Google Scholar
- 6a BARSK~, B. A., AND DEPosE, T.D. Geometric continuity of parametric curves: Three equivalent characterizations. IEEE Comput. Gr. Appl. 9, 6 (Nov. 1989), 60-68. Google Scholar
- 6b BARSKY, B. A., AND DEROSF,, T.D. Geometric continuity of parametric curves: Constructions of geometrically continuous s plines. IEEEComput. Gr. Appl. 10, 1 (Jan. 1990), 60-68. Google Scholar
- 7 BARSKY, B. A., AND DEROSE, T.D. Geometric continuity of parametric curves: Developing the Beta-constraints. Submitted for publication.Google Scholar
- 8 BARTELS, R. H., BEATTY, J. C., AND BARSK~, B. A. An Introduction to Splines for Use in Computer Graphics and Geometric Modeling. Morgan Kaufmann, San Mateo, Calif., 1987. Google Scholar
- 9 BOEHM, W. Curvature continuous curves and surfaces. Comput.-Aided Geom. Des. 2, 4 {Dec. 1985), 313-323. Google Scholar
- 10 BOEHM, W. Rational geometric splines. Comput.-Aided Geom. Des. 4, 1-2 (July 1987), 67-77. Google Scholar
- 11 DUI~IFIIVI, YY ~ OIIIUUbll L;UI-V~ ~:UlU ~Ull~t;~. Ill tZUUllb~blb~- I~1UUU~bI~. ~b~Ul-bbf$II~ U{~ ?~ew ~ iuf$4~, G. Farin, Ed. Society for industrial and Applied Mathematics, Philadelphia, Pa., 1987, pp. 175-184.Google Scholar
- 12 COCHEN,D.,AND LEE,T.M.P.Fast drawing of curves for computer display. In Proceedings of the Spring Joint Computer Conference. AFIPS Press, Montvale, N.J., 1969, pp. 297-307.Google Scholar
- 13 DEROSE, A.D. Geometric c'ontinuity: A parameterization independent measure of continuity for computer aided geometric', design. Ph.D. thesis, Computer Science Div., Dept. of Electrical Engineering and Computer Sciences, Univ. of California, Berkeley, Aug. 1985. (Also available as Tech. Pep. 86-09-04, Dept. of Computer Science, Univ. of Washington, Seattle, 1986; and as Tech. Pep. UCB/CSD 86/255, Computer Science Die., Dept. of Electrical Engineering and Computer Sciences, Univ. of California, Berkeley, 1986.) Google Scholar
- 14 DEPosE, T. D., AND BARSKY, B.A. An intuitive approach to geometric continuity for parametric curves and surfaces. In Proceedings of Graphics Interface '85 (Montreal, May 27-31, 1985). Canadian Information Processing Society, 1985, pp. 343-351. Extended abstract in Proceedings of the International Conference on Computational Geometry and Computer-Aided Design (New Orleans, June 5-8). 1985, pp. '71-75. Revised version published in Computer-Generated Images-- The State of the Art, N. Magnenat-Thaimann and D. Thaimann, Eds. Springer-Verlag, New York, 1985, pp. 159-175. Google Scholar
- 15 DYN, N., EDELMAN, A., AND MICCHELLI, C.A. On locally supported basis functions for the representation of geometrically continuous curves. Res. ~ep. l~l~v, ~iw Thomas J. ~ .. Research Center, Yorktown Heights, N.Y., Sept. 1985.Google Scholar
- 16 DeN, N., AND MICCHELLI, C.A. Piecewise polynomial spaces and geometric continuity of , Re~ ~^~ ~1390, ,u~,~ ~k .. ~ xx~.+~~ ~ ... I. o~~+~ v~.~+ ... ~z~;.h+. ~,~v, Sept. 1985.Google Scholar
- 17 FARIN, G. Visually C2 cubic ~plines. Comput.-Aided Des. 14, 3 (May 1982), 137-139.Google Scholar
- 18 FAR!N, G. Algorithm~ for rational B~ziar curves: Computo-Aided Des. !5, 2 (Mar., 1983), 73-77.Google Scholar
- 19 FAux, I. D., AND PRATT, M.J. Computational Geometry for Design and Manufacture. Ellis Horwood, Chichester, England, 1979. Google Scholar
- 20 FOWLER, A. H., AND WILSON, C.W. Cubic spline, a curve fitting routine. Corp. Pep. Y-1400 (Rev. I.), Union Carbide 1966.Google Scholar
- 21 GOLDMAN, R. N., AND BARSKY, B.A. /~-continuity and its application to rational Beta-splines. In Proceedings of the 4th Computer Graphics Conference (Smolenice, Czechoslovakia, May, 15- 18, 1989). (Also Tech. Pep. UCB/CSD 88/442, Computer Science Division, Electrical Engineering and Computer Sciences Dept., Univ. of California, Berkeley, Aug. 1988.) Google Scholar
- 22 GOLDMAN, R. S., AND BARSKY, B.A. On/~-continuous functions and their application to the construction of geometrically continuous curves and surfaces, in Mathematical Methods in Computer Aided Geometric Design, T. Lyche and L. L. Schumaker, Eds. Academic Press, Boston, 1989, pp. 299-311. (Conference held June 16-22, 1988, in Oslo.) Google Scholar
- 23 GOLDMAN, it. i~., At~t~ wHu~m,~,.~, ~. ~zgeorazc a~pecu~ ot geumevnc cun~znu~y.n ~rlathematical Methods in Computer Aided Geometric Design, T. Lyche and L. L. Schumaker, Eds. Academic Press, Boston, 1989, pp. 313-332. (Conference held June 16-22, 1988, in Oslo.) Google Scholar
- 24 GoodMAN, T.N.T. Properties of Beta-splines. J.Approx. Theory 44,2(June 1985), 132-153.Google Scholar
- 25 GOODMAN, T. N. T., AND UNSWORTH, K. Manipulating shape and producing geometric continuity in Beta-spline curves. IEEE Comput. Gr. Appl. 6, 2 (Feb. 1986), 50-56. (Special Issue on Parametric Curves and Surfaces.)Google Scholar
- 26 HAOEN, H. Geometric spline curves. Comput.-Aided Geom. Des. 2, 1-3 (Sept. 1985), 223-227.Google Scholar
- 27 HAGEN, H. B~zier-curves with curvature and torsion continuity. Rocky Mountain J. Math. 16, 3 (Summer 1986), 629-636.Google Scholar
- 28 HERRON, G. Techniques for visual continuity. In Geometric Modeling: Algorithms and New Trends, G. Farin, Ed. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1987, pp. 163-174.Google Scholar
- 29 HIRSCH, M, W. Differential Topology. Springer-Verlag, New York, 1976.Google Scholar
- 30 HOELLI6, K. Geometric continuity of spline curves and surfaces. Tech. Rep. 645, Computer ocmnce Dept., Univ. of Wisconsin,Ivlacuson," .... Juno.Google Scholar
- 31 JOE, B. Rational Beta-spline curves and surfaces and discrete Beta-splines. Tech. Rep. TR87- 04, Dept. of Computing Science, Univ. of Alberta, Edmonton, Canada, Apr. 1987.Google Scholar
- 32 Programming: ~z.1 .. 1, v..n .. ,.I ~ ~g.;,~.o o.~A ~A Addison-Wesley, Reading, Mass., 1973. Google Scholar
- 33 Lu, Y.-C. Singularity Theory and an Introduction to Catastrophe Theory. Springer-Verlag, New York, !976.Google Scholar
- 34 MANNING, J.R. Continuity conditions for spline curves. Comput. J. 17, 2 (May 1974), 181-186.Google Scholar
- 35 MANOCHA, S., AND BARSKY, B. A. Basis functions for rational continuity. Submitted for publication.Google Scholar
- 36 MANOCHA, D., AND BARSKY, }3. A. Varying the shape parameters of rational continuity. In preparation.Google Scholar
- 37 NIELSON, G.M. Some piecewise polynomial alternatives to splines under tension. In Computer Aided Geometric Design, R. E. Barnhill and R. F. Riesenfeld, Eds. Academic Press, New York, 1974, pp. 209-235.Google Scholar
- 38 PIEGL, L. Representation of quadric primitives by rational polynomials. Comput.-Aided Geom. Des. 2, 1-3 (Sept. 1985), 151-155.Google Scholar
- 39 PIEGL, L. The sphere as a rational B~zier surface. Comput.-Aided Geom. Des. 3, 1 (May 1986), 45-52. Google Scholar
- 40 PIEGL, L. A geometric investigation of the rational B~zier scheme of computer aided design. Comput. Ind. 7, 5 (Oct. 1986), 401-410. Google Scholar
- 41 PIEGL, L. Infinite control points--A method for representing surfaces of revolution using boundary data. IEEE Comput. Graph.Appl 7,3 (mar.1987),485-497/ Google Scholar
- 42 PIEGL, L. On the use of infinite control points in computer aided geometric design. Comput.- Aided Geom. Des. 4, 1-2 (July, 1987), 155-166. Google Scholar
- 43 PIEGL L.,AND TILLER W. Curve and surface constructions for computer aided design using rational B-splines. Comput.-Aided Des., 19, 9 (Nov. 1987), 485-498. Google Scholar
- 44 ROBERTS, L. G. Homogeneous matrix representation and manipulation of N-dimensional constructs. Comput. Display Rev. (May 1965), 1-16. (Also Doc. MS-1405, Lincoln Lab., MIT, Cambridge, Mass., July 1966.)Google Scholar
- 45 SABIN, M.A. Parametric splines in tension. Tech. Rep. VTO/MS/160, British Aircraft Corp., Weybridge, Survey, England, July 1970.Google Scholar
- 46 SPIVAK, M. A Comprehensive Introduction to Differential Geometry, Vol. IIL Publish or Perish, Boston, 1973.Google Scholar
- 47 TILLER, W. Rational B-splines for curve and surface representation. IEEE Comput. Graph. Appl. 3, 6 (Sept. 1983), 61-69.Google Scholar
- 48 VERSPRILLE, K.J. Computer-aided design applications of the rational B-spline approximation form. Ph.D. thesis, Dept. of Computer and Information Science, Syracuse Univ., Syracuse, N.Y., Feb. 1975. Google Scholar
Index Terms
- Rational continuity: parametric, geometric, and Frenet frame continuity of rational curves
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Reviews
Remco C. Veltkamp
The necessary and sufficient constraints for parametric, geometric, and Frenet frame continuity of a rational polynomial curve p u /g u in terms of the components of the associated homogeneous curve p u ,g u are typical subjects in computer-aided geometric design. Continuity for a ra t ional curve has formerly been obtained either by requiring the homogeneous curve to be continuous or by constraints on the components of p u /g u . The authors convincingly demonstrate that the first condition is not always necessary to let the projected curve be continuous. At the end of the paper they also show that their derived constraints on the homogeneous curve provide more freedom than the second condition. The necessary and sufficient constraints on the components of the homogeneous curve are derived from constraints on the components of the projected curve by means of n -jets (the n -jet of a function f ( u ) is f 0 u ,&ldots;,f n u . Hohmeyer and Barsky also derive restrictions on p ( u ) and g ( u ) such that p ( u ) g ( u ) (as opposed to p u /g u ) is parametrically or geometrically continuous. The paper is clearly written, and only a few of the 115 formulae contain typographical errors. The notion of n -jets and many of their properties are presented extensively, which is necessary for the understanding of the proofs of the main results but distracts the reader from the main subject. Especially, the constraints on the projected curve, from which constraints on the homogeneous curve are directly derived, should have been explained (rather than just given). The results of the paper are important because they provide more freedom to design a continuous rational curve than previous results. Be aware, however, of the theoretical nature of the paper; the practice of continuous rational curve construction is not discussed.
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