Standardised co-ordinate geometry applied to affine rational trigonometry of a tetrahedron

TL;DR

This paper introduces a standardized coordinate system for tetrahedra in affine space, enabling simplified analysis of their rational trigonometry invariants through affine transformations and generalized scalar products.

Contribution

It develops a framework combining standardised coordinates with Wildberger's rational trigonometry to facilitate the study of tetrahedral invariants in affine geometry.

Findings

Defined standardised coordinates via the Standard tetrahedron.

Computed trigonometric invariants using the new framework.

Enabled proofs of complex geometric results.

Abstract

The idea of standardised co-ordinates in three-dimensional affine space is defined, by way of the Standard tetrahedron. By performing an affine map on a general tetrahedron, we may replace the study of a general tetrahedron over a specific metrical structure with the study of a specific tetrahedron over a general metrical structure. Using the framework of Wildberger's rational trigonometry as well as the definitions of generalised scalar and vector products in the author's previous work, we will compute the various trigonometric invariants associated to the Standard tetrahedron, with the purpose of proving some more complicated results which are difficult to prove without this mechanism.