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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. Mathematics Subject Classification (2000). Primary 65D07, 65D17; Secondary 68Q25.

The secant map of an immersion sends a pair of points to the direction of the line joining the images of the points under the immersion. The germ of the secant map of a generic codimension-c immersion X: R n → R n+c at the diagonal in the source is a Z 2 stable map-germ R 2n → R n+c−1 in the following cases: (i) c ≥ 2 and (2n, n + c − 1) is a pair of dimensions for which the Z 2 stable germs of rank at least n are dense, and (ii) for generically immersed surfaces (i.e., n = 2 and any c ≥ 1). In the latter surface case the A Z 2 -classification of germs of secant maps at the diagonal is described and it is related to the A-classification of certain singular projections of the surfaces.