# Linkage Mechanisms Governed by Integrable Deformations of Discrete Space Curves

## TL;DR

This paper models spatial linkage mechanisms as discrete space curves with constant torsion, showing their motions are governed by integrable semi-discrete mKdV equations, revealing new insights into their geometric and dynamic properties.

## Contribution

It introduces a novel geometric framework linking spatial linkages to integrable curve deformations governed by semi-discrete mKdV equations.

## Key findings

- Linkage motions correspond to integrable curve deformations.
- The model includes the Kaleidocycle as a special case.
- The deformation preserves torsion and arc length.

## 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 spacial 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 Bricard6R 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 $\mathbb{R}^3$ with a 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 equations, where infinitesimally the motion of each vertex is confined in the osculating plane.