Approximation by Conic Splines |
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Authors: | Sunayana Ghosh Sylvain Petitjean Gert Vegter |
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Institution: | (1) Department of Mathematics and Computing Science, University of Groningen, PO Box 407, NL-9700 AK Groningen, The Netherlands;(2) LORIA-INRIA, Campus Scientifique, BP 239, F-54506 Vandoeuvre cedex, France |
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Abstract: | We show that the complexity of a parabolic or conic spline approximating a sufficiently smooth curve with non-vanishing curvature
to within Hausdorff distance ɛ is c
1ɛ−1/4 + O(1), if the spline consists of parabolic arcs, and c
2ɛ−1/5 + O(1), if it is composed of general conic arcs of varying type. The constants c
1 and c
2 are expressed in the Euclidean and affine curvature of the curve. We also show that the Hausdorff distance between a curve
and an optimal conic arc tangent at its endpoints is increasing with its arc length, provided the affine curvature along the
arc is monotone. This property yields a simple bisection algorithm for the computation of an optimal parabolic or conic spline.
The research of SG and GV was partially supported by grant 6413 of the European Commission to the IST-2002 FET-Open project
Algorithms for Complex Shapes in the Sixth Framework Program. |
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Keywords: | Approximation splines conics Hausdorff distance complexity differential geometry affine curvature affine spiral |
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