Approximation by Conic Splines

We show that the complexity of a parabolic or conic spline approximating a sufficiently smooth curve with non-vanishing curvature to within Hausdorff distance ε is c1ε^−1/4 + O(1), if the spline consists of parabolic arcs, and c2ε^−1/5 + O(1), if it is composed of general conic arcs of varying type. The constants c1 and c2 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.

Complexity of Approximation by Conic Splines (Extended Abstract)

In this paper we show that the complexity, i.e., the number of elements, of a parabolic or conic spline approximating a sufficiently smooth curve with non-vanishing curvature to within Hausdorff distance ε is c1ε^−1/4 + O(1), or c2ε^−1/5 + O(1), respectively. The constants c1 and c2 are expressed in the Euclidean and affine curvature of the curve. We also prove 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. We use this property in a simple bisection algorithm for computing an optimal parabolic or conic spline.