The complexity of robot motion planning
The complexity of robot motion planning
The molecule problem: determining conformation from pairwise distances
The molecule problem: determining conformation from pairwise distances
Conditions for unique graph realizations
SIAM Journal on Computing
Handbook of discrete and computational geometry
Different bounds on the different Betti numbers of semi-algebraic sets
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
A user-based evaluation of skeletal animation techniques in graph interaction
APVis '05 proceedings of the 2005 Asia-Pacific symposium on Information visualisation - Volume 45
Similarity queries in data bases using metric distances – from modeling semantics to its maintenance
EUROCAST'05 Proceedings of the 10th international conference on Computer Aided Systems Theory
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(MATH) Rigid frameworks in some Euclidian space are embedded graphs having a unique local realization (up to Euclidian motions) for the given edge lengths, although globally they may have several. We study first the number of distinct planar embeddings of rigid graphs with n vertices. We show that, modulo planar rigid motions, this number is at most $2n-4\choose n-2 \approx 4n. We also exhibit several families which realize lower bounds of the order of 2n, 2.21n and 2.88n.(MATH) For the upper bound we use techniques from complex algebraic geometry, based on the (projective) Cayley-Menger variety CM 2,n(C)\subset P_n\choose 2-1(C)$ 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(CM 2,n) zero-dimensional components, and one finds this degree to be D 2,n =\frac122n-4\choose n-2$. The lower bounds are related to inductive constructions of minimally rigid graphs via Henneberg sequences.(MATH) The same approach works in higher dimensions. In particular we show that it leads to an upper bound of 2 D^3,n= \frac2^n-3n-2n-6\choosen-3$ for the number of spatial embeddings with generic edge lengths of the $1$-skeleton of a simplicial polyhedron, up to rigid motions.