Lupaşq-Bézier曲线的显式递归生成

摘要

为了得到具有更好性质的Lupaşq-Bézier曲线的递归求值算法，通过应用Pascal-type关系和重新参数化，构造具有显式矩阵表示的de Casteljau算法，并得到具有对称性质的Lupaşq-Bézier曲线。首先，利用Pascal-type关系构造具有显式矩阵表示的de Casteljau算法，该算法具有经典Bézier曲线的de Casteljau算法的3个性质；然后，通过重新参数化调整Lupaşq-Bézier曲线上点的分布，得到具有对称性质的Lupaşq-Bernstein基函数和Lupaşq-Bézier曲线，给出重新参数化后 Lupaş q-Bézier 曲线的一种矩阵累乘的递归生成方法。另外，从应用角度给出了用一条 Lupaşq-Bézier曲线逼近2条光滑拼接的Bézier曲线的数值实例，进而验证了文中算法的有效性。%To obtain Lupaşq-Bézier curves by recursive evaluation algorithms with better properties, new de Casteljau algorithms and Lupaşq-Bézier curves with symmetry are constructed by means of Pascal-type formula and reparameterization. A new de Casteljau algorithm with explicit matrix representation is constr-ucted by applying Pascal-type formula, and the algorithm shares three properties with de Casteljau algorithm of classical Bézier curves. Lupaşq-Bernstein basis functions and Lupaşq-Bézier curves with symmetry are gained from reparameterization, moreover, Lupaşq-Bézier curves reparameterized can be generated by mul-tiply bidiagonal matrices successively on control polygon. In addition, numerical examples of using one Lupaşq-Bézier curve to approximate two blending Bézier curves are presented as a simple application of de Casteljau algorithm with explicit matrix representation and the effectiveness of the algorithm is verified.