Universality theorems for linkages in homogeneous surfaces

Abstract

Abstract. A mechanical linkage is a mechanism made of rigid rods linked together by flexible joints, in which some vertices are fixed and others may move. The partial configuration space of a linkage is the set of all the possible positions of a subset of the vertices. We characterize the possible partial configuration spaces of linkages in the (Lorentz-)Minkowski plane, in the hyperbolic plane and in the sphere. We also give a proof of a differential universality theorem in the Minkowski plane and in the hyperbolic plane: for any compact manifold M , there is a linkage whose configuration space is diffeomorphic to the disjoint union of a finite number of copies of M . In the Minkowski plane, it is also true for any manifold M which is the interior of a compact manifold with boundary. CHAPTER 1 Introduction and generalitiesA mechanical linkage is a mechanism made of rigid rods linked together by flexible joints. Mathematically, we consider a linkage as a marked graph: lengths are assigned to the edges, and some vertices are pinned down while others may move.A realization of a linkage L in a manifold M is a mapping which sends each vertex of the graph to a point of M, respecting the lengths of the edges. The configuration space Conf M (L) is the set of all realizations. Intuitively, the configuration space is the set of all the possible states of the mechanical linkage. This supposes, classically, the ambient manifold M to have a Riemannian structure: thus the configuration space may be seen as the space of "isometric immersions" of the metric graph L in M.Here we will always deal with (non-trivially) marked connected graphs, that is, a non-empty set of vertices have fixed realizations (in fact, when M is homogeneous, considering a linkage without fixed vertices only adds a translation factor to the configuration space). Hence, our configurations spaces will be compact even if M is not compact, but rather complete. Some historical backgroundMost existing studies deal with the special case where M is the Euclidean plane and some with the higher dimensional Euclidean case (see for instance [Far08] Universality theorems. When M is the Euclidean plane E 2 , a configuration space is an algebraic set. This set is smooth for a generic length structure on the underlying graph.Universality theorems tend to state that, playing with mechanisms, we get any algebraic set of R n , and any manifold, as a configuration space! In contrast, it is a hard task to understand the topology or geometry of the configuration space of a given mechanism, even for a simple one.Universality theorems have been announced in the ambient manifold E 2 by Thurston in oral lectures, and then proved by Kapovich and Millson in [KM02]. They have been proved in E n by King [Kin98], and in RP 2 and in the 2-sphere by Mnëv (see [Mnë88] and [KM02]). It is our aim in the present article to prove them in the cases of: the hyperbolic plane H 2 , the sphere S 2 and the (Lorentz-)Minkowski plane M. These are simply connected homogeneous pseudo-Riemannian surfa...

Elementary proofs of Kempe universality

“…For example it can be shown that any compact connected smooth manifold is diffeomorphic to a connected component of the configuration space of a pinned planar linkage. Related developments in this direction may be found in Kourganoff [18]. 4.2.…”

Anosov Geodesic Flows, Billiards and Linkages

“…In 2002, Kapovich and Millson [KM02] showed that any compact algebraic set B ⊆ R 2n is exactly the partial configuration space of some linkage, that is, the set of the possible positions of a subset of the vertices; moreover, if B a smooth submanifold of R 2n , each connected component of the whole configuration space may be required to be smooth and diffeomorphic to B (see also [Kou14] for more details). Thus, there is a linkage and a subset of the vertices W such that the partial configuration space of W is A: each component of the configuration space of this linkage, with masses 1 for the vertices in W and 0 for the others, is isometric to the algebraic set A endowed with the metric induced by R 2n , which is itself isometric to (M, h).…”