Introduction to Clifford's Geometric Algebra

TL;DR

This tutorial introduces geometric algebra, a unifying mathematical framework that simplifies the representation of geometric transformations and has diverse applications across physics, robotics, computer vision, and neural computing.

Contribution

It provides a comprehensive introduction to geometric algebra, illustrating its concepts with concrete examples and demonstrating its utility in various scientific and engineering fields.

Findings

Geometric algebra unifies multiple branches of physics and engineering.

It enables efficient representation of geometric transformations.

Applications include robotics, signal processing, and neural computing.

Abstract

Geometric algebra was initiated by W.K. Clifford over 130 years ago. It unifies all branches of physics, and has found rich applications in robotics, signal processing, ray tracing, virtual reality, computer vision, vector field processing, tracking, geographic information systems and neural computing. This tutorial explains the basics of geometric algebra, with concrete examples of the plane, of 3D space, of spacetime, and the popular conformal model. Geometric algebras are ideal to represent geometric transformations in the general framework of Clifford groups (also called versor or Lipschitz groups). Geometric (algebra based) calculus allows, e.g., to optimize learning algorithms of Clifford neurons, etc. Keywords: Hypercomplex algebra, hypercomplex analysis, geometry, science, engineering.