AI 中文总结
该研究利用广义Chapple–Euler关系证明中心二次曲线与单位圆构成3-庞塞勒对的充要条件，通过马登定理实现相关庞塞勒族的对称参数化，推导几何不变量并扩展马登定理至留数理论框架。
AI 中文摘要
利用广义的Chapple–Euler关系，我们首先证明：具有焦点 $a_1,a_2\in\mathbb C$ 的中心二次曲线与单位圆构成3-庞塞勒对当且仅当其长轴长度为 $|1-\overline{a_1}a_2|$，该刻画对椭圆和双曲线情形均适用。我们的主要结果表明，马登定理可给出这类庞塞勒族中每一个的对称参数化：若 $z_1,z_2,z_3\in\mathbb T$ 是外切三角形的顶点，则其初等对称函数可通过焦点 $a_1,a_2$ 及单位模参数 $\lambda \in \mathbb T$ 显式表达。由此直接导出经典的三次Blaschke乘积方程，且该方程无需焦点位于单位圆盘内仍成立。该参数化为推导外切三角形族的几何不变量提供了统一框架：我们得到了与垂心和边长相关的不变量，并用简短证明给出经典定理——抛物线的三条切线构成的三角形的外接圆过其焦点。最后，我们为有限个极点的部分分式建立了高次对称参数化，导出广义默比乌斯乘积方程，其关联留数满足扩展经典马登权重的单位分解恒等式，从而将马登定理置于更广泛的留数理论框架中，并为庞塞勒几何中的高次类比提供了可能。
英文摘要
The classical Blaschke-product approach provides an elegant description of triangles inscribed in a circle and circumscribed about an ellipse. Motivated by the Siebeck--Marden theorem, we derive a symmetric parametrization for cyclic $p$-gons circumscribed about a Siebeck--Marden curve of class $p-1$, recovering the Blaschke product parametrization as the triangular case. We then use this parametrization to establish several geometric invariance results, including a characterization of when the sum of the squares of all sides and diagonals is invariant. We revisit Cayley's criterion for 3- and 4-Poncelet pairs and obtain a complete classification of the associated central conics solely in terms of their foci. In particular, we show that the formulas for the lengths of the major axis obtained under the assumption on the foci of the conic lying inside the circumcircle remain valid when one focus or both foci lie outside the circumcircle, thereby extending the classical theory from ellipses to all admissible central conics. Several geometric properties of the associated cyclic quadrilaterals are also obtained.
Comments27 pages, 7 figures
