Contributions to Four-Position Theory with Relative Rotations

TL;DR

This paper explores the geometric properties of four spatial displacements with pure rotations between neighbors, identifying specific loci related to points, planes, and lines under these motions.

Contribution

It introduces new geometric loci associated with four-position theory involving relative rotations, expanding understanding of spatial displacement configurations.

Findings

Locus of points with homologous images on a circle

Locus of planes tangent to a cone of revolution

Locus of lines forming a skew quadrilateral on a hyperboloid of revolution

Abstract

We consider the geometry of four spatial displacements, arranged in cyclic order, such that the relative motion between neighbouring displacements is a pure rotation. We compute the locus of points whose homologous images lie on a circle, the locus of oriented planes whose homologous images are tangent to a cone of revolution, and the locus of oriented lines whose homologous images form a skew quadrilateral on a hyperboloid of revolution.