Discrete Gliding Along Principal Curves

TL;DR

This paper explores discrete motions called rotation nets, showing their connection to principal contact element nets and constructing motions with multiple trajectories, advancing understanding of discrete Euclidean displacements.

Contribution

It establishes the relationship between rotation nets and principal contact element nets and constructs discrete rotating motions with multiple trajectories, extending to higher dimensions.

Findings

Every principal contact element net can be realized as a trajectory of a discrete rotating motion.

Discrete rotating motions with two non-parallel principal contact element net trajectories are constructible.

Such rotation nets can be extended to higher dimensions.

Abstract

We consider $n$-dimensional discrete motions such that any two neighbouring positions correspond in a pure rotation ("rotating motions"). In the Study quadric model of Euclidean displacements these motions correspond to quadrilateral nets with edges contained in the Study quadric ("rotation nets"). The main focus of our investigation lies on the relation between rotation nets and discrete principal contact element nets. We show that every principal contact element net occurs in infinitely many ways as trajectory of a discrete rotating motion (a discrete gliding motion on the underlying surface). Moreover, we construct discrete rotating motions with two non-parallel principal contact element net trajectories. Rotation nets with this property can be consistently extended to higher dimensions.