High-order approximation of conic sections by quadratic splines

Abstract

Given a segment of a conic section in the form of a rational Bézier curve, a quadratic spline approximation is constructed and an explicit error bound is derived. The convergence order of the error bound is shown to be O( h 4) which is optimal, and the spline curve is both C 1 and G 2. The approximation method is very efficient as it is based on local Hermite interpolation and subdivision. The approximation method and error bound are also applied to an important subclass of rational biquadratic surfaces which includes the sphere, ellipsoid, torus, cone and cylinder.