Poncelet-Jacobi双中心多项式的新不变量
New Invariants of Poncelet-Jacobi Bicentric Polygons
摘要：
在两个圆之间相交的庞斯莱多边形的一维族被称为双中心族。利用椭圆函数和Liouville定理，我们证明了（i）这个族具有不变的内角余弦和和（ii）关于族的极限点的踏板多边形具有不变的周长.有趣的是，（i）和（ii）也是椭圆台球N-S的性质。此外，由于踏板多边形在（ii）是相同的椭圆台球N-π的反转相对于一个焦点为中心的圆，一个重要的推论是，（iii）椭圆台球焦点反转N-π有恒定的周长。有趣的是，这些也保存了它们的余弦之和（除了N=4的情况）。
英文摘要：
The 1d family of Poncelet polygons interscribed between two circles is known as the Bicentric family. Using elliptic functions and Liouville's theorem, we show (i) that this family has invariant sum of internal angle cosines and (ii) that the pedal polygons with respect to the family's limiting points have invariant perimeter. Interestingly, both (i) and (ii) are also properties of elliptic billiard N-periodics. Furthermore, since the pedal polygons in (ii) are identical to inversions of elliptic billiard N-periodics with respect to a focus-centered circle, an important corollary is that (iii) elliptic billiard focus-inversive N-gons have constant perimeter. Interestingly, these also conserve their sum of cosines (except for the N=4 case).
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