Gábor Hegedüs

Factorization of rational curves in the study quadric

Abstract For every generic rational curve C in the group of Euclidean displacements we construct a linkage such that the constrained motion of one of the links is exactly C. Our construction is based on the factorization of polynomials over dual quaternions.

The Theory of Bonds: A New Method for the Analysis of Linkages

In this paper we introduce a new technique, based on dual quaternions, for the analysis of closed linkages with revolute joints: the theory of bonds. The bond structure comprises a lot of information on closed revolute chains with a one-parametric mobility. We demonstrate the usefulness of bond theory by giving a new and transparent proof for the well-known classification of overconstrained 5R linkages.

Construction of Overconstrained Linkages by Factorization of Rational Motions

We prove that for any sufficiently generic rational curve C of degree n in the group of Euclidean displacements, there exists an overconstrained spatial linkage with revolute joints whose linkage graph is the 1-skeleton of the n-dimensional hypercube such that the constrained motion of one of the links is exactly C. The synthesizing algorithm is based on the factorization of polynomials over the dual quaternions. The linkage contains n! open nR chains, so that low degree examples include Bennett’s mechanisms and are related to overconstrained 5R and 6R chains.

The theory of bonds II

In this paper, we study closed linkages with six rotational joints that allow a one-dimensional set of motions. We prove that the genus of the configuration curve of such a linkage is at most five, and give a complete classification of the linkages with a configuration curve of genus four or five. The classification contains new families.

Four-Pose Synthesis of Angle-Symmetric 6R Linkages

We use the recently introduced factorization theory of motion polynomials over the dual quaternions for the synthesis of closed kinematic loops with six revolute joints that visit four prescribed poses. Our approach admits either no or a one-parametric family of solutions. We suggest strategies for picking good solutions from this family.

Bond Theory and Closed 5R Linkages

We present bond theory as a new means for the analysis of overconstrained closed linkages with revolute joints. Intuitively, bonds are special points in the complex configuration curve. They exhibit discrete properties which can be visualized in bond diagrams and allow to read off directly certain properties such as the degrees of relative motions, or the special geometry of consecutive revolute axes. As an application we sketch a classification of overconstrained 5R linkages.

THE BOUNDARY VOLUME OF A LATTICE POLYTOPE

For a d-dimensional convex lattice polytope P, a formula for the boundary volume vol(δP) is derived in terms of the number of boundary lattice points on the first [d/2] dilations of P. As an application we give a necessary and sufficient condition for a polytope to be reflexive, and derive formulae for the f-vector of a smooth polytope in dimensions 3, 4, and 5. We also give applications to reflexive order polytopes, and to the Birkhoff polytope.