

Geometric Algebra

D. Hildenbrand , D. Fontijne, C. Perwass, L. Dorst 

G.4 [Mathematical Software]: Algorithm design and analysis, Efficiency
I.3.7 [Computer Graphics]: Animation
Early in the development of computer graphics it was realized that projective geometry is very well suitable to represent points and transformations. Now we can realize that another change of paradigm is lying ahead of us. It is based on the socalled Geometric Algebra.
Maybe you already use quaternions or lie algebra in additon to the wellknown vector algebra. You may ask "Why should I use another mathematical system ?".
We will emphasize in this half day tutorial that Geometric Algebra
 is a unified language for a lot of mathematical systems used in Computer Graphics
 can be used in an easy and geometrically intuitive way in Computer Graphics.
We will focus on the (5D) Conformal Geometric Algebra. It is an extention of the 4D projective geometric algebra. E.g. spheres and circles are simply represented by algebraic objects. To represent a circle you only have to take two spheres ( or a sphere and a plane ) and to intersect them with help of the meet operation. E.g. the inverse kinematics of a robot can be computed in an easy way.
Only with help of this kind of geometric intuitve operations you are able to compute the joint angles of a robot in order to reach its goal location.
In this tutorial we will give an overview of Geometric Algebra and its application to computer graphics. First of all, we want to motivate the topic and give insights into some applications.
In particular, the Conformal Geometric Algebra with its socalled 'conformal model' of 3dimensional Euclidean geometry will be introduced. In this model, Euclidean objects and their interactions will be explored and visualized interactively.
With help of the conformal model we will describe animations and motions. It will be shown how it can be used quite advantageously to treat this kind of computer graphics applications. We will give some basic visual examples and describe rigid body motions and their interpolations. We will focus on the inverse kinematics and dynamics of kinematic chains in order to describe motions of robots and human figures.
At the university of Amsterdam a ray tracer was developed in order to compare different geometric approaches from the implementation and performance point of view. Compared to linear algebra, the richer mathematical language of GA leads to more work for implementing the algebra, but less work for implementing the application. We discuss the issues in implementing a numerical geometric algebra package for a language like C++. We compare various existing implementations and look at their performance. We conclude with future implementation methods like SIMD hardware suitable for GA and generative programming.
During the tutorial only the most fundamental mathematical aspects of Geometric Algebra will be presented. This is possible, since most aspects of Geometric Algebra can be understood with geometric intuition. The actual mathematical 'inner workings' of the algebra will be detailed in an accompanying script that also contains many visual examples from the presentations.
The tutorial will be rounded off by an outlook into possible future applications of Geometric Algebra in computer graphics.
Intelligent Autonomous Systems Informatics
Institute Faculty of Sciences
University of Amsterdam
Amsterdam, The Netherlands
email: fontijne@science.uva.nl
Phone : 3120525 7511
Fax : 3120525 7490Daniel Fontijne is a scientific programmer and PhD student at the University of Amsterdam. His main goal is to integrate geometric algebra into various programming environments and languages in ways that are both efficient and usable. He has earned an MSc with distinction in Artificial Intelligence. Previously, he presented a course on geometric algebra at the Game Developers Conference 2003 and published a tutorial on geometric algebra in IEEE Computer Graphics and Applications.
University of Technology Darmstadt, Germany
Fraunhoferstr. 5
64283 Darmstadt
email:dietmar.hildenbrand@gris.informatik.tudarmstadt.de
Phone: +49 6151 155 667
Fax: +49 6151 155 669
URL:http://www.dgm.informatik.tudarmstadt.de/staff/dietmar
Dietmar Hildenbrand is a researcher and PhD student with the Interactive Graphics Systems group in Darmstadt, Germany. He holds a Masters degree in Computer Science. His main research interest lies in describing animations with the help of Geometric Algebra. He prepared a tutorial on Geometric Algebra together with C. Perwass for the DAGM 2003 conference.
university of Kiel, Germany
Institut fuer Informatik,
Olshausenstr. 40,
24098 Kiel, Germany
email: christian@perwass.de
Tel.: +49 431 8807548,
Fax: +49 431 8807550,
URL : www.perwass.deChristian Perwass is a postdoctoral researcher and assistant teacher at the Cognitive Systems Group of the University of Kiel, Germany. His main research interest lies in the application of Geometric Algebra to Computer Vision and related fields. He holds a MSci degree in Physics and a PhD in applications of Geometric Algebra in Computer Vision.
He regularly gives a course on Geometric Algebra at the University of Kiel, and has presented a tutorial on Geometric Algebra together with D. Hildenbrand at the DAGM 2003 conference. He recently published an awardwinning paper on the application of Geometric Algebra in artificial neural networks and is the author of a visualization and teaching software program for Geometric Algebra.
Intelligent Autonomous Systems
Informatics Institute
Faculty of Sciences University of Amsterdam
Amsterdam, The Netherlands
email: leo@science.uva.nl
Phone : 3120525 7511
Fax : 3120525 7490
Leo Dorst is an assistant professor at the Informatics Institute of the University of Amsterdam. His research interests include geometric algebra and its applications to computer science. He has an MSc and PhD in the applied physics of computer vision.
Previously he presented lectures at various specialist conferences on geometric algebra and copresented geometric algebra courses at SIGGRAPH 2000 and 2001. He recently copublished three tutorials on geometric algebra in IEEE Computer Graphics and Applications.
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Page last edited on7/22/2004 .