By interpreting planar polynomial curves as complex-valued functions of a real parameter, an inner product, norm, metric function, and the notion of orthogonality may be defined for such curves. This approach is applied to the complex pre-image polynomials that generate planar Pythagorean-hodograph (PH) curves, to facilitate the implementation of bounded modifications of them that preserve their PH nature. The problems of bounded modifications under the constraint of fixed curve end points and end tangent directions, and of increasing the arc length of a PH curve by a prescribed amount, are also addressed.

翻译：将平面多项式曲线解释为实参数的复值函数后，可为此类曲线定义内积、范数、度量函数及正交性概念。该方法被应用于生成平面Pythagorean-hodograph（PH）曲线的复原像多项式，以实现在保持PH曲线本质属性的前提下对其进行有界修改。本文还研究了在固定曲线端点及端点切线方向约束下的有界修改问题，以及将PH曲线弧长增加指定长度的问题。