Approximating conic sections by constrained Bézier curves of arbitrary degree

Abstract

In this paper, an algorithm for approximating conic sections by constrained Bézier curves of arbitrary degree is proposed. First, using the eigenvalues of recurrence equations and the method of undetermined coefficients, some exact integral formulas for the product of two Bernstein basis functions and the denominator of rational quadratic form expressing conic section are given. Then, using the least squares method, a matrix-based representation of the control points of the optimal Bézier approximation curve is deduced. This algorithm yields an explicit, arbitrary-degree Bézier approximation of conic sections which has function value and derivatives at the endpoints that match the function value and the derivatives of the conic section up to second order and is optimal in the L2 norm. To reduce error, the method can be combined with a curve subdivision scheme. Computational examples are presented to validate the feasibility and effectiveness of the algorithm for a whole curve or its part generated by a subdivision.