Abstract Given a segment of a conic section in the form of a rational quadratic Bézier curve and any positive odd integer n, a geometric Hermite interpolant, with 2 n contacts, counting multiplicity, is presented. This leads to a G n−1 spline approximation having an approximation order of O( h 2 n ). A bound on the Hausdorff error of the Hermite interpolant is provided. Both the interpolation and error bound are extended to an important subclass of rational biquadratic Bézier surfaces. For low n, the approximation provides a method for converting the so-called analytic curves and surfaces used in CAGD to polynomial spline form with very small error.