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中文摘要：
为了更加方便清晰地应用复形式的有理deCasteljau算法和细分算法,通过研究一次复有理Bézier曲线的最优参数化问题,提出2种最优参数化方法——代数方法和几何方法.代数方法借助直接的代数运算推导曲线在Mbius变换下的重新参数化,使得这种参数化在L2范数下最接近于弧长参数化;而几何方法从一次复有理Bézier曲线的内在几何性质出发,直接求得曲线在Mbius变换下的最优参数化,进而揭示曲线最优参数化的本质.另外,从应用角度给出了用一次复有理Bézier曲线插值3个给定点的公式.实验结果表明,在最优参数化后,曲线上的等参数点分布更加均匀,因而拥有更强的实用性.
英文摘要：
To apply the complex version of the rational de Casteljau algorithm and the subdivision algorithm in a convenient manner,optimal parameterization of complex rational Bézier curves of degree one is studied.Both algebraic and geometric methods to derive the optimal parameterization are presented.The algebraic method is to deduce the curve＇s reparameterization under the Mbius transformations by direct algebraic computation,so that the new parameterization is the closest to the arc parameterization under the L2 norm.The geometric method is to directly deduce the optimal parameterization under the Mbius transformations by applying the intrinsic geometric properties of the complex rational Bézier curves of degree one,and then the essence of optimal parameterization is obtained.In addition,a formula for interpolating three given points by a complex rational Bézier curve of degree one is presented as an application of reparameterization.Numerical examples show that the iso-parametric points on the curve are uniform after optimal parameterization.
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