Point Configurations and Cayley-Menger Varieties

Abstract

Equivalence classes of $n$-point configurations in Euclidean, Hermitian, and quaternionic spaces are related, respectively, to classical determinantal varieties of symmetric, general, and skew-symmetric bilinear forms. Cayley-Menger varieties arise in the Euclidean case, and have relevance for mechanical linkages, polygon spaces and rigidity theory. Applications include upper bounds for realizations of planar Laman graphs with prescribed edge-lengths and examples of special Lagrangians in Calabi-Yau manifolds.

Cited by in corpus (9)

• On the maximal number of real embeddings of minimally rigid graphs in ${\mathbb{R}}^2$, ${\mathbb{R}}^3$ and $S^2$
• The algebro-geometric study of range maps
• On the maximal number of real embeddings of spatial minimally rigid graphs
• Counting Euclidean embeddings of rigid graphs
• Determining Generic Point Configurations From Unlabeled Path or Loop Lengths
• On tangents to quadric surfaces
• Linear Symmetries of the Unsquared Measurement Variety
• Algebraic methods for counting Euclidean embeddings of rigid graphs
• Mixed Volume Techniques for Embeddings of Laman Graphs