Abstract
Based on the support function, a well-known tool in the theory of convex sets, a new measure of a “distance” between two planar parametric curves is introduced and its main properties are established. The measure is invariant under affine transformations and depends not only on the position vectors but also on the tangent vectors of the curves. An algorithm to compute the optimal Bézier approximant to a given curve is derived.
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
Boor, C. de, Höllig, K., Sabin, M.: High accuracy geometric Hermite interpolation. Comput. Aided Geom. Des. 4, 269–278 (1987).
Degen, W. L. F.: Best approximation of parametric curves by splines. In: Mathematical methods in computer aided geometric design II (Lyche, T., Schumaker, L. L., eds.), pp. 171–184. New York: Academic Press, 1992.
Degen, W. L. F.: Best approximation of parametric curves by splines. In: Geometric modeling (Farin, G., Hagen, H., Noltemeier, H., eds.), pp. 59–74. Wien New York: Springer, 1993.
Degen, W. L. F.: High accurate rational approximation of parametric curves. Comput. Aided Geom. Des. 10, 293–313 (1993).
Dokken, T., Dæhlen, M., Lyche, T., Mørken, K.: Good approximation of circles by curvature-continuous Bézier curves. Comput. Aided Geom. Des. 33–42 (1990).
Eisele, E.: Chebyshev approximation of plane curves by splines. J. Approx. Theory 76, 133–148 (1994).
Goult, R.: Parametric curve and surface approximation. In: The mathematics of surfaces III (Handscomb, D. C., ed.), pp. 331–346. Oxford: Clarendon Press, 1989.
Koch, J.: Algorithms for geometric Hermite interpolation. In: Approximation theory VIII, Vol. 1, Approximation and Interpolation (Chui, Ch. K., Schumaker, L. L., eds.), pp. 281–286. Singapore: World Scientific, 1995.
Leichtweiß, K.: Konvexe Mengen. Berlin: Deutscher Verlag der Wissenschaften, 1980.
Morken, K., Scherer, K.: A general framework for high accuracy parametric interpolation. Math. Comput. 66, 237–260(1997).
Schabak, R.: Rational geometric curve interpolation. In: Mathematical methods in computer aided geometric design II (Lyche, T., Schumaker, L. L., eds.), pp. 517–535. New York: Academic Press, 1992.
Schneider, R.: Convex bodies: the Brunn-Minkowski theory. Cambridge: Cambridge University Press, 1993.
Valentine, F.: Convex sets. New York: McGraw-Hill, 1964.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer-Verlag Wien
About this paper
Cite this paper
Degen, W.L.F. (1998). Approximation of Curves by a Measure of Shape. In: Farin, G., Bieri, H., Brunnett, G., De Rose, T. (eds) Geometric Modelling. Computing Supplement, vol 13. Springer, Vienna. https://doi.org/10.1007/978-3-7091-6444-0_8
Download citation
DOI: https://doi.org/10.1007/978-3-7091-6444-0_8
Publisher Name: Springer, Vienna
Print ISBN: 978-3-211-83207-3
Online ISBN: 978-3-7091-6444-0
eBook Packages: Springer Book Archive