Planar linkages following a prescribed motion

Abstract

Designing mechanical devices, called linkages, that draw a given plane curve has been a topic that interested engineers and mathematicians for hundreds of years, and recently also computer scientists. Already in 1876, Kempe proposed a procedure for solving the problem in full generality, but his constructions tend to be extremely complicated. We provide a novel algorithm that produces much simpler linkages, but works only for parametric curves. Our approach is to transform the problem into a factorization task over some noncommutative algebra. We show how to compute such a factorization, and how to use it to construct a linkage tracing a given curve.

Kempe’s Universality Theorem for Rational Space Curves

“…Kempe's Theorem talks about algebraic curves and it is natural to ask for simplifications in case of rational curves. This was done recently in [10] where the authors constructed scissor-like linkages to draw rational planar curves. Their construction is based on the factorization of certain polynomials over a non-commutative ring that describe the motion of one of the links.…”

“…The upper bounds on the number of links and joints for curves of degree d reduce dramatically to 3d + 2 and 9 2 d + 1, respectively. In this article, we extend the ideas of [10] to rational space curves. Our aim is to provide a construction that works for all rational space curves and, at the same time, to reduce the number of links and joints as far as possible.…”

Factorization results for left polynomials in some associative real algebras: State of the art, applications, and open questions

“…One example related to a rational version of Kempe's Universality Theorem is depicted in Figure 3. Any rational planar or spatial curve (an ellipse in Figure 3) can be drawn by a scissor like linkage whose number of joints is linear in the curve degree [7,8]. Figure 3.…”

“…The factorization theory of polynomials over division rings has been developed half a century ago in [1,2]. It gained new attention in recent years because relations to mechanism science were unveiled [3,4,5,6,7,8,9,10,11]. Quaternion polynomials parameterize rational spherical motions.…”

The Geometry of Quadratic Quaternion Polynomials in Euclidean and Non-euclidean Planes

“…Assume at first that H 1 is contained in the interior of N . There exists a suitable isometry of hyperbolic geometry that maps H 1 to [1]. Moreover, we pick an initial position of the four-bar linkage where H 2 lies on the line spanned by [i] and [j] and apply a suitable linear re-parametrization that ensures h 2 + h 2 = 0.…”