Linkage mechanisms governed by integrable deformations of discrete space curves

Abstract

A linkage mechanism consists of rigid bodies assembled by joints which can be used to translate and transfer motion from one form in one place to another. In this paper, we are particularly interested in a family of spatial linkage mechanisms which consist of n-copies of a rigid body joined together by hinges to form a ring. Each hinge joint has its own axis of revolution and rigid bodies joined to it can be freely rotated around the axis. The family includes the famous threefold symmetric Bricard 6R linkage, also known as the Kaleidocycle, which exhibits a characteristic "turning-over" motion. We can model such a linkage as a discrete closed curve in R 3 of constant torsion up to sign. Then, its motion is described as the deformation of the curve preserving torsion and arc length. We describe certain motions of this object that are governed by the semi-discrete mKdV and sine-Gordon equations, where infinitesimally the motion of each vertex is confined in the osculating plane.

Infinitesimal Darboux transformation and semi-discrete mKdV equation

“…Then it was the result of [32,33,37,38] that revealed a relationship between various semi-discrete and discrete analogues of the KdV and mKdV equations (see, for example, [24,26]), and the compatibility condition associated with discretization of the moving frames. In fact, by considering the isoperimetric deformation of discrete space curves and the corresponding moving frames, [35] recovered a particular form of the semi-discrete potential mKdV equation that can be found in [43] in the context of the transformation and permutability of solutions to the smooth mKdV equation, suggesting a close relationship between the moving frames approach and transformation theory.…”

“…Preliminaries -isoperimetric deformation. We briefly review the isoperimetric deformation of discrete plane curves as considered in [32,35] (see also [18,19,27,37]), primarily to introduce notations. Let x : Σ → C be a planar curve which is regular, that is, on any three consecutive vertices, det (x n − x n−1 x n+1 − x n ) = 0.…”

Infinitesimal Darboux transformation and semi-discrete MKDV equation

“…Then it was the result of [35,36,43,44] that revealed a relationship between various semi-discrete and discrete analogues of the KdV and mKdV equations (see, for example, [28,29]), and the compatibility condition associated with discretization of the moving frames. In fact, by considering certain motions of discrete space curves and the corresponding moving frames, [38] recovered a particular form of the semi-discrete potential mKdV equation that can be found in [54] in the context of the transformation and permutability of solutions to the smooth mKdV equation, suggesting a close relationship between the moving frames approach and transformation theory.…”

Invertible Paradoxic Loop Structures for Transformable Design

“…The invertible cube can also be seen as a special dimensioned kaleidocycle , which are closed chains of hinged tetrahedra [SW77]. These invertible rings trace back to the graphic designer Walker [Wal67], and they are still a topic of recent research [KKP19]. The mathematical loop with the most impact in the art/design community is the Möbius strip [Pic06], whose kaleidocyclic realization was presented recently [SF19].…”