We classify planar polynomial Pythagorean-hodograph curves of any degree with respect to Euclidean similarities. We also analyze possible global shapes of the Pythagorean-hodograph curves of degree four and five and describe all possible configurations of their singular points. This description results in a classification of Pythagorean-hodograph quartics and quintics with respect to the homeomorphisms of the plane. We also describe all the Pythagorean-hodograph curves of any degree up to five which exhibit central or axial symmetry.

中文翻译：

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毕达哥拉斯平面波谱仪曲线的分类

我们对欧几里得相似度的任意程度的平面多项式勾股-hodograph曲线进行分类。我们还分析了度数为4和5的勾股勾线图曲线的可能整体形状，并描述了它们奇异点的所有可能构型。该描述导致关于平面的同胚性的勾股四面体四次方和五次方分类。我们还描述了显示中心或轴向对称性的所有不超过五度的勾股勾线图。