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Idzhad Kh. Sabitov - Solution of polyhedra
| IDZHAD KH. SABITOV, Faculty of Mechanics and Mathematics, Moscow State University, 119899 Moscow, Russia |
| Solution of polyhedra |
By analogy with the chapter of elementary mathematics named ``the
solution of triangles'' we can propose an idea for finding all metric
characteristics of a polyhedron based on the knowledge of its metric
and combinatorial structure. The essential moment of the proposed
approach is the generalized Heron's formule for volumes of isometric
polyhedra established in [1]. Namely we affirm that the volumes of all
isometric polyhedra with a fixed combinatorial simplicial structure
Kand given lengths (l) of its edges are roots of a polynomial
equation
whose coefficients are, in one's turn, polynomials in edges' lengths with some numerical coefficients depending only on the combinatorial structure K. For the construction of a such equation one can indicate some corresponding algorithm. It turns out moreover that for the polyhedra in general position we can find only a finite number of values of its diagonals so we have a finite algorithm for the construction of isometric polyhedra. Some simple cases of this algorihm are realized on the computer.
References
1. I. Kh. Sabitov, Fund. i Prikl. Mat. 2(1996), 1235-1246.
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