inria-00188456, version 1

## Approximation by conic splines

Sunayana Ghosh 1, Sylvain Petitjean () 2, Gert Vegter 1

Mathematics in Computer Science 1, 1 (2007) 39-69

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.

• Domain : Computer Science/Computational Geometry
• Keywords : Approximation – splines – conics – Hausdorff distance – complexity – differential geometry – affine curvature – affine spiral

• inria-00188456, version 1
• oai:hal.inria.fr:inria-00188456
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• Submitted on: Thursday, 30 June 2011 17:53:09
• Updated on: Friday, 1 July 2011 09:55:31