Planar polynomial curves have rational offset curves, if they are either Pythagorean-hodograph (PH) or indirect Pythagorean-hodograph (iPH) curves. In this paper, we derive an algebraic and two geometric characterizations for planar quartic iPH curves. The characterizations are given in terms of quantities related to the Bézier control polygon of the curve, and naturally extend to quartic and cubic PH and quadratic iPH curves.

中文翻译：

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一类有理偏移的平面四次曲线的代数和几何特征

如果平面多项式曲线是毕达哥拉斯-hodograph（PH）或间接毕达哥拉斯-hodograph（iPH）曲线，则它们具有有理偏移曲线。在本文中，我们推导了平面四次iPH曲线的代数和两个几何特征。这些特征是根据与曲线的Bézier控制多边形有关的数量给出的，并且自然地扩展到四次和三次PH和二次iPH曲线。