Euclidean Geometric Objects in the Clifford Geometric Algebra of {Origin, 3-Space, Infinity}

TL;DR

This paper explores the conformal model of Euclidean space using Clifford geometric algebra, providing explicit constructions and parametrizations to extract 3D geometric information such as positions, orientations, and radii.

Contribution

It introduces detailed methods for modeling 3D Euclidean objects within the Clifford algebra framework, enabling precise extraction of geometric data.

Findings

Explicit parametrizations of Euclidean objects in Clifford algebra

Methods to extract positions, orientations, and radii from models

Demonstration of the conformal model's utility in 3D geometry

Abstract

This paper concentrates on the homogeneous (conformal) model of Euclidean space (Horosphere) with subspaces that intuitively correspond to Euclidean geometric objects in three dimensions. Mathematical details of the construction and (useful) parametrizations of the 3D Euclidean object models are explicitly demonstrated in order to show how 3D Euclidean information on positions, orientations and radii can be extracted.