计算单形最小包围和外切圆柱的代数方法
Algebraic methods for computing smallest enclosing and circumscribing cylinders of simplices
摘要：
我们提供了一个代数框架来计算欧几里得空间$\E^n$中单形的最小封闭和最小外切圆柱。说明了在$\mathbb{E}^3$中对最小封闭圆柱的计算可归结为对最小外接圆柱的计算。我们改进了现有的多项式公式计算局部极端外切圆柱$\E^3$和表现出的代数程度可以进一步减少的单形的子类。此外，我们推广这些有效的配方的$n$维的情况下，并提供当地极值的数量上的界限。利用初等不变量理论，证明了正则单形任意局部极值外切圆柱的方向向量的结构性结果。
英文摘要：
We provide an algebraic framework to compute smallest enclosing and smallest circumscribing cylinders of simplices in Euclidean space $\E^n$. Explicitly, the computation of a smallest enclosing cylinder in $\mathbb{E}^3$ is reduced to the computation of a smallest circumscribing cylinder. We improve existing polynomial formulations to compute the locally extreme circumscribing cylinders in $\E^3$ and exhibit subclasses of simplices where the algebraic degrees can be further reduced. Moreover, we generalize these efficient formulations to the $n$-dimensional case and provide bounds on the number of local extrema. Using elementary invariant theory, we prove structural results on the direction vectors of any locally extreme circumscribing cylinder for regular simplices.
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