Rigidity with few locations
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RIGIDITY WITH FEW LOCATIONS
BY
Karim Adiprasito and Eran Nevo Einstein Institute of Mathematics, The Hebrew University of Jerusalem Givat Ram, Jerusalem 91904, Israel e-mail: [email protected], [email protected]
ABSTRACT
Graphs triangulating the 2-sphere are generically rigid in 3-space, due to Gluck–Dehn–Alexandrov–Cauchy. We show there is a finite subset A in 3-space so that the vertices of each graph G as above can be mapped into A to make the resulted embedding of G infinitesimally rigid. This assertion extends to the triangulations of any fixed compact connected surface, where the upper bound obtained on the size of A increases with the genus. The assertion fails, namely no such finite A exists, for the larger family of all graphs that are generically rigid in 3-space and even in the plane.
1. Introduction A theorem of Dehn asserts that the 1-skeleton of every simplicial convex 3polytope is infinitesimally rigid [Deh16]. Combined with the Steinitz theorem, this gives Gluck’s result that the 1-skeleton of any simplicial 2-sphere is generically rigid in R3 [Glu75], i.e., the locus of realizations that are not infinitesimally rigid is of codimension one in the configuration space of all possible locations. See [Con93] and [Pak] for further references and discussion. We ask the following question: How generic does the embedding of a generically rigid graph need to be to guarantee that it is infinitesimally rigid? We give a natural precise meaning to this meta question, and partially answer it for various families of graphs, including the one mentioned above. Received August 26, 2018 and in revised form October 16, 2019
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K. ADIPRASITO AND E. NEVO
Isr. J. Math.
Let us first recall some notions pertaining to infinitesimal rigidity: An embedding of a graph G = (V, E) into Rd is any map f : V → Rd such that f (V ) affinely spans Rd ; it defines a realization of the edges in E by segments via linear extension, and this realization is called the framework f (G). A motion of f (G) is any assignment of velocity vectors a : V → Rd that satisfies (1)
a(v) − a(u), f (v) − f (u) = 0
for every edge uv ∈ E. A motion a is trivial if the relation (1) is satisfied for every pair of vertices; otherwise a is non-trivial. The framework f (G) is infinitesimally rigid if all its motions are trivial. Equivalently, (1) says that the velocities preserve infinitesimally the distance along an embedded edge, and if (1) applies to all pairs of vertices then the velocities necessarily correspond to a rigid motion of the entire space. We now arrive at the central definition of this note, quantifying the genericity of the embedding needed for an infinitesimally rigid embedding. Definition 1.1: Let F be a family of graphs. We say F is d-rigid with c-locations if there exists a set A ⊆ Rd of cardinality c such that for any graph G = (V (G), E(G)) ∈ F there exists a map f : V (G) → A such that the framework f (G) is infinitesimally rigid. Denote by cd (F ) the minimal such c. We are interested in the question whether a given infinite famil
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