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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

\begin{displaymath}Q(V)=V^{2N}+a_1(l)V^{2N-2}+\cdots +a_N(l)=0, \end{displaymath}

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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Next: Peter Schmitt - The Up: Discrete Geometry / Géométrie Previous: Konstantin Rybnikov - On
 
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