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.