This paper deals with Pythagorean hodograph curves of the spaces of algebraic trigonometric functions and trigonometric polynomials, \(\text {span}\left \{1,t,\left \{{\cos \limits } \left ({kt}\right ), {\sin \limits } \left ({kt}\right )\right \}_{k=1}^{m}\right \} \) and \( \text {span}\left \{1,\left \{{\cos \limits } \left ({kt}\right ), {\sin \limits } \left ({kt}\right )\right \}_{k=1}^{m}\right \}, \) respectively. First, we propose a general characterization of planar PH curves in these spaces. Next, we consider the particular cases m = 1 and m = 2. For each of them, we give the general form of the control polygon of the PH curves, the implicit relations defining these curves, and their geometrical interpretations. Some examples and particular cases complete this study.

中文翻译：

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三次和五次代数三角平面PH曲线的表征和广泛研究

本文处理代数三角函数和三角多项式空间的勾股勾勒曲线图，\（\ text {span} \ left \ {1，t，\ left \ {{\ cos \ limits} \ left（{kt} \ right），{\ sin \ limits} \ left（{kt} \ right \\ right \} _ {k = 1} ^ {m} \ right \} \）和\（\ text {span} \ left \ { 1，\ left \ {{\ cos \ limits} \ left（{kt} \ right），{\ sin \ limits} \ left（{kt} \ right）\ right \} _ {k = 1} ^ {m } \ right \}，\）。首先，我们提出了这些空间中平面PH曲线的一般特征。接下来，我们考虑特殊情况m = 1和m=2。对于它们中的每一个，我们都给出PH曲线的控制多边形的一般形式，定义这些曲线的隐式关系以及它们的几何解释。一些示例和特殊情况可以完成本研究。