Document Type
Article
Publication Date
2-2004
Publication Title
Discrete & Computational Geometry
Abstract
Rigid frameworks in some Euclidean space are embedded graphs having a unique local realization (up to Euclidean motions) for the given edge lengths, although globally they may have several. We study the number of distinct planar embeddings of minimally rigid graphs with $n$ vertices. We show that, modulo planar rigid motions, this number is at most ${{2n-4}\choose {n-2}} \approx 4^n$. We also exhibit several families which realize lower bounds of the order of $2^n$, $2.21^n$ and $2.28^n$. For the upper bound we use techniques from complex algebraic geometry, based on the (projective) Cayley--Menger variety ${\it CM}^{2,n}(C)\subset P_{{{n}\choose {2}}-1}(C)$ over the complex numbers $C$. In this context, point configurations are represented by coordinates given by squared distances between all pairs of points. Sectioning the variety with $2n-4$ hyperplanes yields at most $deg({\it CM}^{2,n})$ zero-dimensional components, and one finds this degree to be $D^{2,n}=\frac{1}{2}{{2n-4}\choose {n-2}}$. The lower bounds are related to inductive constructions of minimally rigid graphs via Henneberg sequences. The same approach works in higher dimensions. In particular, we show that it leads to an upper bound of $2 D^{3,n}= {({2^{n-3}}/({n-2}})){{2n-6}\choose{n-3}}$ for the number of spatial embeddings with generic edge lengths of the $1$-skeleton of a simplicial polyhedron, up to rigid motions. Our technique can also be adapted to the non-Euclidean case.
Volume
31
Issue
2
First Page
287
Last Page
303
DOI
10.1007/s00454-003-2902-0
ISSN
1432-0444
Recommended Citation
Borcea, Ciprian and Streinu, Ileana, "On the Number of Embeddings of Minimally Rigid Graphs" (2004). Computer Science: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/csc_facpubs/17
Comments
Author’s submitted manuscript.