柔性多面体理论中的代数与分析
Algebra versus analysis in the theory of flexible polyhedra
摘要：
用完全不同的方法证明了柔性多面体理论的两个基本定理：R。亚历山大使用分析，即斯托克斯定理，以证明总中曲率保持不变，在弯曲，而I. Kh。Sabitov使用代数，即理论的resultants，以证明定向体积保持不变，在弯曲。我们表明，没有这些方法可以用来证明这两个定理。作为一个副产品，我们证明了总中曲率的任何多面体在欧几里德3-空间是不是一个代数函数的边长。
英文摘要：
Two basic theorems of the theory of flexible polyhedra were proven by completely different methods: R. Alexander used analysis, namely, the Stokes theorem, to prove that the total mean curvature remains constant during the flex, while I.Kh. Sabitov used algebra, namely, the theory of resultants, to prove that the oriented volume remains constant during the flex. We show that none of these methods can be used to prove the both theorems. As a by-product, we prove that the total mean curvature of any polyhedron in the Euclidean 3-space is not an algebraic function of its edge lengths.
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