A Unified Framework of Elementary Geometric Transformation Representation

TL;DR

This paper introduces stereohomology, a unified framework extending projective homology, to represent various elementary geometric transformations as homogeneous matrices, enabling coordinate-independent transformation representation.

Contribution

It proposes stereohomology as an extension of Desargues theorem, unifying multiple geometric transformations into a single matrix framework.

Findings

All elementary geometric transformations can be represented as homogeneous matrices.

Transformations are independent of coordinate system choices.

The framework includes reflection, translation, projection, and scaling.

Abstract

As an extension of projective homology, stereohomology is proposed via an extension of Desargues theorem and the extended Desargues configuration. Geometric transformations such as reflection, translation, central symmetry, central projection, parallel projection, shearing, central dilation, scaling, and so on are all included in stereohomology and represented as Householder-Chen elementary matrices. Hence all these geometric transformations are called elementary. This makes it possible to represent these elementary geometric transformations in homogeneous square matrices independent of a particular choice of coordinate system.