基于约束Jacobi基的多项式反函数逼近及应用

Inversion Approximation for Polynomials by the Constrained Jacobi Basis and its Application

求解多项式反函数是CAGD中的一个基本问题.提出一种带端点Ck约束的反函数逼近算法.利用约束Jacobi基作为有效工具,推导了它与Bernstein基的转换公式,采用Bernstein多项式的升阶、乘积、积分与组合运算,给出了求解反函数系数的具体算法.该算法稳定、简易,克服了以往计算反函数的系数时每次逼近系数需全部重新计算的缺陷.最后通过具体逼近实例验证了文中算法的正确性和有效性,同时给出了它在PH曲线准弧长参数化中的应用.

To solve the inverse function of polynomial is a fundamental problem in CAGD. An algorithm about approximating the inverse function with C^k constrains is proposed. By using the constrained Jacobi basis and a derived transformation formula for it to Bernstein basis, and using the degree elevation, arithmetic and composition algorithms for Bernstein polynomials, the specific method for solving the coefficients of inverse function is given. The approximation method is convenient and steady. Moreover, the defect that the corresponding coefficients must be recalculated when approximating every inverse function one by one was overcame. Finally, the experimental results show that the approximation methods are correctness and effective. As an application, generating quasi arclength parameterization of PH curves is also discussed.