论文标题
Euler和大惊小怪定理的较高维度版本
Higher dimensional versions of theorems of Euler and Fuss
论文作者
论文摘要
我们介绍了Euler和大惊小怪的经典结果的更高维度版本，这都是著名的Poncelet Porism的特殊情况。我们的结果涉及多面体，特别是简单，并行型和交叉多型，并刻在给定的椭圆形中，并限制在另一个。陈述和证明使用线性代数的语言。椭圆形的一个是单位球，另一个也以原点为中心。令$ a $为正对称矩阵，将外椭圆形带到内部。如果$ trace a = 1 $，则在正交组$ o（n）$和此类标记的简单组合之间存在两次培养。同样，如果$ trace a^2 = 1 $，则有一个并行的家族和交叉多型家族，也由$ o（n）$索引。
We present higher dimensional versions of the classical results of Euler and Fuss, both of which are special cases of the celebrated Poncelet porism. Our results concern polytopes, specifically simplices, parallelotopes and cross polytopes, inscribed in a given ellipsoid and circumscribed to another. The statements and proofs use the language of linear algebra. Without loss, one of the ellipsoids is the unit sphere and the other one is also centered at the origin. Let $A$ be the positive symmetric matrix taking the outer ellipsoid to the inner one. If $trace A = 1$, there exists a bijection between the orthogonal group $O(n)$ and the set of such labeled simplices. Similarly, if $trace A^2 = 1$, there are families of parallelotopes and of cross polytopes, also indexed by $O(n)$.
