On transformations and graphic methods of algebraically 3 dimensional force, velocity and displacement systems

TL;DR

This paper explores how three-dimensional force, velocity, and displacement systems in engineering can be represented and analyzed using projective geometry and linear transformations, linking mechanical systems to geometric methods.

Contribution

It introduces a method to factorize 3D vector spaces of forces and motions, connecting them to points and lines in projective plane, and extends previous results with new insights.

Findings

Factorization of 3D force, velocity, and displacement spaces

Connection between projective transformations and linear maps of these spaces

Extension of previous results in the geometric analysis of mechanical systems

Abstract

In engineering practice one often encounters planar problems, where the corresponding vector space of forces, velocities or (infinitesimal) displacements is three dimensional. This paper shows how these spaces can be factorized, such that the arising equivalence classes correspond to points and lines of action of the forces / velocities / displacements in the (projective) plane. It is shown how the study of projective transformations and dualities of planar mechanical systems is closely related to the study of linear maps of these spaces. A few past results are analysed and sometimes extended to show the power of this description.