Approximating conic sections by constrained Bezier curves of arbitrary degree

作者：Hu, Qian-Qian^*

来源：Journal of Computational and Applied Mathematics, 2012, 236(11): 2813-2821.

DOI：10.1016/j.cam.2012.01.017

• Conic sections
• Approximation
• Bezier curves
• Arbitrary degree
• Least squares method

In this paper, an algorithm for approximating conic sections by constrained Bezier 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 Bezier approximation curve is deduced. This algorithm yields an explicit, arbitrary-degree Bezier 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 L-2 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.

• 出版日期2012-5
• 单位浙江工商大学