Hands on Geometric Algebra
A Geometry and computational unifying framework for representing and transforming geometric objects
at Asian Conference on Pattern Recognition - ACPR 2019 - Auckland, New Zealand, 26-29 November 2019
Welcome to the geometric algebra tutorial
Geometric algebra provides convenient and intuitive tools to represent, transform, and intersect geometric objects. Using only points and one operator, called wedge, it is possible to construct and transform and intersect geometric objects ranging from circles, planes to quadric and quartic surfaces. Furthermore, the objects manipulated get rid of the coordinates, thus resulting in a very powerful and computational geometric object-oriented language. Finally, all the operators are defined in any dimension thus making much easier the extension of methods to n-dimension. These features combined with the computationnaly efficient feature of these frameworks is the based to explore both the discretization and discrete transformations of surfaces.
The aim of this tutorial is to give a general panorama of geometric algebra and to present some applications in discrete geometry and computer vision. The tutorial is thought for and not limited to students, teachers and researchers who work mainly in either discrete geometry or computer vision. The tutorial will be enriched with examples given using geometric algebra and discrete geometry software.
I will introduce geometric algebra with the basic ideas behind it. We will discuss the history, then the main operators of Geometric Algebra.
Compass ruler algebra
This part will present two frameworks that allows to get rid of the matrix and to intuitively represent a wide range of geometric objects. We will explain how it is possible to capture geometric properties of a geometric object in a single algebraic entity. This will be our base for defining transformations over geometric objects.
Geometric transformations, discrete geometry and discrete transformations
This part is divided into three
- the first point focuses on the efficient and intuitive representation of geometric transformations including the representation of exponential versors.
- second contribution is dedicated to the intuitive discretization of geometric objects by means of geometric algebra. Topological properties of the resulting objects will be discussed. Finally, a method to fit geometric entities will be presented.
- Finally, I will present discrete transformations by means of inversions and reflections and it is defined for any dimension. This results in efficient definitions of discrete transformations.
Efficient implementation of geometric algebra
Stéphane Breuils is a postdoc working at the National Institute of Informatics in Tokyo, Japan. He got a PhD in computer science at LIGM, University of Paris-Est in 2018. His main topic is geometric algebra and its applications, including in discrete geometry.