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Approximation by conic splines

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 $\varepsilon$ is $c_1\varepsilon^{-\frac{1}{4}} + O(1)$, if the spline consists of parabolic arcs, and $c_2\varepsilon^{-\frac{1}{5}} + O(1)$, 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.
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Submitted on : Thursday, June 30, 2011 - 5:53:09 PM
Last modification on : Wednesday, February 2, 2022 - 3:51:47 PM
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Sunayana Ghosh, Sylvain Petitjean, Gert Vegter. Approximation by conic splines. Mathematics in Computer Science, Springer, 2007, 1 (1), pp.39-69. ⟨10.1007/s11786-007-0004-8⟩. ⟨inria-00188456⟩

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