On the definition of combinatorially p-parametric polyhedra
by
Igor Maksimov
Moscow State University
Coauthors: Idjad Sabitov (Moscow State University)

We treat the problem of rigidity as one on the local uniqueness of isometric realization of polyhedral metrics in R³ (up to motion of polyhedron as a rigid solid).

More precisely: let K be a simplicial complex homeomorphe to a 2-dimensional compact manifold and provided by a polyhedral metric, that is having some given lengths of 1-simplices. The problem is to find a continuous map P:K +AFw-to R³, linear on each simplex, which realizes prescribed metric on the polyhedron P(K).

One important aspect of considered problem is to determine the degree of ambiguity in the selection of such a map. This is expressed by the number of parameters which are needed to be added in order to get the unique map. It turns out that the correct definition needs very accuracy and detailed explanations.

As the first step we offer to estimate the number of parameters determined only by the combinatorial structure K of a polyhedron P contrary to the possible contribution of some external particularities of disposition of the polyhedron in the space.

Let n and E be correspondingly the number of vertices of K and the set of its 1-simplices (i, j), +AH4-1+AFw-le i, j+AFw-le n, the lengths of which are designed as l_ij, +AH4-(i, j)+AFw-in E. Let (x_i, y_i, z_i), +AH4-1+AFw-le i +AFw-le n, be the coordinates in R³ of vertices of a polyhedron P(K) isometric to K, and let (d_km), +AH4-(k, m) +AFw-notin E be the set of lengths of the polyhedron's diagonals (k, m)+AFw-notin E.

We say that P(K) is a generic polyhedron if there are no three vertices of P which would be situated on a straight line.

Since a polyhedron in R³ is uniquely (up to motions) determined by the complete set of its edges' and diagonals' lengths we have two equivalent ways for the description of polyhedra: 1) to give the coordinates (x_i, y_i, z_i)+AFw-in R³, +AH4-1+AFw-le i+AFw-le n or 2) to give the lengths (l_ij, d_km)+AFw-in R^N, +AH4-(i, j)+AFw-in E, +AH4-(k, m)+AFw-notin E, +AH4-N = +AFw-displaystyle+AFw-frac n(n-1)2.

Let D(K) +AFw-subset R^N be the set of N-tuples (l_ij, d_km) each of which has a realization in R³ as the edges' and diagonals' lengths of some generic polyhedron P(K). Since we know the combinatorial structure of P we can write the equations which determine the domain D(K) using for this Cayley-Menger determinantes. These equations depend only on the polyhedron's combinatorial structure K and they motivate the following

Definition. A polyhedron P:K +AFw-to R³ is called combinatorially p-parametric if one can indicate p diagonals (k_s, m_s), +AH4-1+AFw-le s+AFw-le p, such that the intersection of D(K) with the set of planes l_ij=Const, +AH4-d_{ks, ms}=Const, for any choice of Const is either empty or consists of a finite number of points.

In other words this definition means that for a combinatorially p-parametric polyhedron we can choose p diagonals such that their lengths being fixed the polyhedron becomes continuosly rigid (inflexible) for any choice of its metric, if it is in the general position in the above sense. For example the combinatorial structure of any suspension is such that it is combinatorially 1-parametric.

The work of Maksimov was partially supported by the RFBR grant 99-01-00867. The work of Sabitov was partially supported by the RFBR-INTAS grant No. IR-97-1778

Date received: February 21, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadw-38.