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Arc Length of Rational Bezier Curves and Use for CAD Reparametrization

Authors: Maharavo Randrianarivony


The length  of a given rational B'ezier curve is efficiently estimated. Since a rational B'ezier function is nonlinear, it is usually impossible to evaluate its length exactly. The length is approximated by using subdivision and the accuracy of the approximation n is investigated. In order to improve the efficiency, adaptivity is used with some length estimator. A rigorous theoretical analysis of the rate of convergence of n to  is given. The required number of subdivisions to attain a prescribed accuracy is also analyzed. An application to CAD parametrization is briefly described. Numerical results are reported to supplement the theory.

Keywords: Adaptivity, Length, Parametrization, Rational Bezier

Digital Object Identifier (DOI):

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[1] G. Brunnett, "Geometric design with trimmed surfaces", Computing Supplementum, vol. 10, pp. 101-115, 1995.
[2] C. de Boor, A practical guide to splines, New York: Springer Verlag, 1978.
[3] G. Farin, Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide, Boston: Academic Press, 1997.
[4] O. Figueiredo, J. Reveilles and R. Hersch, "Digitization of B'ezier curves and patches using discrete geometry", in Proc. 8th Int. Conf. Discrete Geometry for Computer Imagery, Marne-la-Vall'ee, 1999, pp. 388-398.
[5] M. Floater, "Chordal cubic spline interpolation is fourth order accurate", IMA J. Numer. Anal., vol. 26, pp. 25-33, 2006.
[6] T. Hain, "Rapid termination evaluation for recursive subdivision of B'ezier curves", in: Proc. Int. Conf. on Image Science, Systems, and Technology, Las Vegas, 2002, pp. 323-328.
[7] H. Prautzsch, W. Boehm and M. Paluszny, B'ezier and B-Spline techniques, Berlin: Springer Verlag, 2002.
[8] M. Randrianarivony, "Geometric processing of CAD data and meshes as input of integral equation solvers". Ph.D. dissertation, Dept. Comput. Science, Chemnitz University of Technology, Chemnitz, Germany, 2006.
[9] M. Randrianarivony and G. Brunnett, "Molecular surface decomposition using geometric techniques", in Proc. Conf. Bildverarbeitung f¨ur die Medizine, Berlin, 2008, pp. 197-201.
[10] M. Randrianarivony and G. Brunnett, "Preparation of CAD and Molecular Surfaces for Meshfree Solvers", in Proc. Int. Workshop Meshfree Methods for PDE, Bonn, 2007, pp. 231-245.
[11] J. Roulier, "Specifying the arc length of B'ezier curves", Computer Aided Geometric Design, Vol. 10, pp. 25-56, 1993.
[12] H. Seidel, "Polar forms for geometrically continuous spline curves of arbitrary degree", ACM Trans. Graph., Vol. 12, pp. 1-34, 1993.
[13] M. Walter and A. Fournier, "Approximate arc length parametrization", in Proc. 9th Brazilian Symposium on Computer Graphics and Image Processing, Brazil, 1996, pp. 143-150