Abstract
Expressing complex curves with simple parametric curve segments is widely used in computer graphics, CAD and so on. This paper applies rational quadratic B-spline curves to give a global C 1 continuous approximation to a large class of plane parametric curves including rational parametric curves. Its application in approximate implicitization is also explored. The approximated parametric curve is first divided into intrinsic triangle convex segments which can be efficiently approximated with rational quadratic Bézier curves. With this approximation, we keep the convexity and the cusp (sharp) points of the approximated curve with simple computations. High accuracy approximation is achieved with a small number of quadratic segments. Experimental results are given to demonstrate the operation and efficiency of the algorithm.
Similar content being viewed by others
References
Ahn, Y.: Conic approximation of plane curves. Comput. Aided Des. 33(12), 867–872 (2001)
Bajaj, C., Xu, G.: Piecewise rational approximation of real algebraic curves. J. Comput. Math. 15(1), 55–71 (1997)
Bazarra, M., Sherali, H., Shetty, C.: Non-linear Programming: Theory and Algorithms. Wiley, New York (1993)
Blomgren, R., Fuhr, R.: Algorithm to convert between rational b-spline and rational Bézier representation of curves and surfaces. Boeing Commercial Airplane Company, Renton, WA (16) (1982)
de Boor, C., Höllig, K., Sabin, M.: High accuracy geometric hermite interpolation. Comput. Aided Geom. Des. 4(4), 269–278 (1987)
Chang, G., Sederberg, T.: Over and Over Again. The Mathematical Association of America, Washington DC (1998)
Cho, W., Maekawa, T., Patrikalakis, N.: Topologically reliable approximation of composite bezier curves. Comput. Aided Geom. Des. 13(6), 497–520 (1996)
Chuang, J., Hoffmann, C.: On local implicit approximation and its application. ACM Trans. Graphics 8(4), 298–324 (1989)
Degen, W.: High accurate rational approximation of parametric curves. Comput. Aided Geom. Des. 10(3–4), 293–313 (1993)
Dokken, T.: Approximate implicitization. In: Lyche, T., Schumaker, L.L. (eds.) Mathematical Methods in CAGD. Vanderbilt University Press, Nashville (2001)
Farin, G.: Curvature continuity and offsets for piecewise conics. ACM Trans. Graphics 8(2), 89–99 (1989)
Gao, X., Li, M.: Rational quadratic approximation to plane real algebraic curves. Comput. Aided Geom. Des. 21(8), 805–828 (2004)
Hu, J., Pavlidis, T.: Function plotting using conic splines. IEEE Comput. Graphics Appl. 11(1), 89–94 (1991)
Johnson, J.: Algorithms for polynomial real root isolation. In: B.F. Caviness, J.R. Johnson (eds.) Quantifier Elimination and Cylindrical Algebraic Decomposition, pp. 269–299. Springer, Berlin Heidelberg New York (1998)
Kotsireas, I., Lau, E.: Implicitization of polynomial curves. In: Computer Mathematics. World Scientific, Singapore (2003)
Lee, E.: The rational Bézier representation for conics. In: G. Farin (ed.) Geometric Modeling: Algorithm and New Trends, pp. 3–19. SIAM, Philadelphia (1985)
Li, Y., Cripps, R.: Identification of inflection points and cusps on rational curves. Comput. Aided Geom. Des. 14(5), 491–497 (1997)
Montaudouin, Y., Tiller, W., Vold, H.: Application of power series in computational geometry. Comput. Aided Des. 18(10), 93–108 (1986)
Park, H.: Choosing nodes and knots in closed b-spline curve interpolation to a point data. Comput. Aided Des. 33(13), 967–974 (2001)
Piegl, L.: Interactive data interpolation by rational bézier curves. IEEE Comput. Graphics Appl. 7(4), 45–58 (1987)
Pottmann, H.: Locally controllable conic splines with curvature continuity. ACM Trans. Graphics 10(4), 366–377 (1991)
Pottmann, H., Leopoldseder, S., Hofer, M.: Approximation with active b-spline curves and surfaces. In: Coquillart, S., Shum, H.Y. (eds.) Pacific Graphics 2002 Proceedings. IEEE Computer Society, Los Alamitos, CA (2002)
Pratt, V.: Techniques for conic splines. ACM Trans. Graphics 19(3), 151–159 (1985)
Quan, L.: Conic reconstruction and correspondence from two views. IEEE Trans. Patt. Analy. Mach. Intell. 18(2), 151–160 (1996)
Sánchez-Reyes, J., Chacón, J.: Polynomial approximation to clothoids via s-power series. Comput. Aided Des. 35(14), 1305–1313 (2003)
Sederberg, T., Zheng, J., Klimaszewski, K., Dokken, T.: Approximate implicitization using monoid curves and surfaces. Graphic. Models Images Process. 61(4), 177–198 (1999)
Shalaby, M., Juttler, B., Schicho, J.: c 1 spline implicitization of planar curves. In: Automated Deduction in Geometry, pp. 161–177 (2002)
Sherbrooke, E., Patrikalakis, N.: Computation of the solution of non-linear polynomial systems. Comput. Aided Geom. Des. 10(5), 379–405 (1993)
Wang, G., Sederberg, T., Chen, F.: On the convergence of polynomial approximation of rational functions. J. Approx. Theory 89(3), 267–288 (1997)
Wang, W., Pottmann, H., Liu, Y.: Fitting b-spline curves to point clouds by squared distance minimization. ACM Trans. Graphics 25(2), 214–238 (2006)
Yang, X.: Curve fitting and fairing using conic splines. Comput. Aided Des. 36(5), 461–472 (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Li, M., Gao, XS. & Chou, SC. Quadratic approximation to plane parametric curves and its application in approximate implicitization . Visual Comput 22, 906–917 (2006). https://doi.org/10.1007/s00371-006-0075-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00371-006-0075-6