Abstract
This paper discusses a solution to the problem of how to approximate an algebraic curve of any degree with piecewise parametric curves. The approximation can be performed using any differentiable parametric curve form. An implementation using rational cubic Bézier curves results in a G 2 piecewise approximation which experiences better than O(h 7) convergence. If the algebraic curve happens to be a cubic curve of genus zero, a parametric curve can be found which exactly fits the given algebraic curve to within floating point precision. An error bound is developed which is much tighter than previously published bounds.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Abhyankar, S.S. and Bajaj, C. (1987a), Automatic rational parametrization of curves and surfaces IE: Cubics andcubicoids, Computer-Aided Design, 19,9,499–502.
Abhyankar, S.S. and Bajaj, C. (1987b), Automatic rational parametrization of curves and surfaces III: algebraic plane curves, Computer Science Technical Report, CSD-TR-619, Purdue University.
Arnon, D.S. (1983), Topologically reliable display of algebraic curves, Computer Graphics, 17, 3, 219–227.
Bajaj, C.L., Hoffmann, CM., Hopcroft, J.E., and Lynch, R.E.(1987), Tracing surface intersections, Technical Report CSD-TR-728, Computer Science Department, Purdue University.
de Boor, C, Höllig, K. and Sabin, M.A.(1987), High accuracy geometric Hermite interpolation, Computer Aided Geometric Design, 4,269–278.
Chuang, J.H. and Hoffmann, CM. (1988), On local implicit approximation and its applications, CSD-TR-812, Purdue University.
Davis, P.J. (1975), Interpolation and Approximation, Dover Publications, Inc., New York.
de Montaudouin, Y., Tiller, W. and Vold, H. (1986), Applications of power series in computational geometry, Computer-Aided Design 18, 514–524.
DeRose, T.D. (1988), Composing Bézier simplices, ACM Transactions on Graphics, 7, 198–221.
Farin, G. (1986), Triangular Bernstein-Bézier patches, Computer Aided Geometric Design, 3, 83–127.
Farouki, R.T. (1986), The characterization of parametric surface sections, Computer Vision, Graphics and Image Processing, 33,209–236.
Farouki, R.T. and Rajan, V.T.(1988), Algorithms for polynomials in Bernstein form, Computer Aided Geometric Design, 5,1–26.
Sabin, M.A. (1968), Conditions for continuity of surface normals between adjacent parametric patches, Tech. Report, British Aircraft Corporation, Ltd.
Sederberg, T.W.(1984), Piecewise algebraic curves, Computer Aided Geometric Design, 1, 241–255.
Sederberg, T.W., Anderson, D.C. and Goldman, R.N.(1984), Implicit representation of parametric curves and surfaces, Computer Vision, Graphics, and Image Processing, 28, 72–84.
Sederberg, T.W. (1988), An algorithm for algebraic curve intersection, to appear in Computer-Aided Design.
Waggenspack, W.N. (1987), Parametric curve approximations for surface intersections, Ph.D. Thesis, Department of Mechanical Engineering, Purdue University.
Waggenspack, W.N. and Anderson, D.C. (1988), Piecewise approximation to algebraic curves, to appear in Computer Aided Geometric Design.
Watkins, M. A. and Worsey, A. J.( 1988), Degree reduction of Bézier curves, Computer-Aided Design, 20, 7, 398–405.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1989 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Sederberg, T.W., Zhao, J., Zundel, A.K. (1989). Approximate Parametrization of Algebraic Curves. In: Straßer, W., Seidel, HP. (eds) Theory and Practice of Geometric Modeling. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-61542-9_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-61542-9_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-64866-3
Online ISBN: 978-3-642-61542-9
eBook Packages: Springer Book Archive