ASCI course a20, part Geometric Algebra, April 19-20, 2007
Interactive Downloads for the Lectures
The lectures on geometric algebra will involve real-time demonstrations and experimentation. You are recommended to bring your laptop for those days (or share one with another attendee). You will need to download from
www.geometricalgebra.net
both GAviewer and the example illustrations in figures.zip, and install them as specified. (These will also be made available during the course, but we will obviously save time if you do this before.)
Lecture Slides
I did most on the blackboard, but the slides of the introduction are here.
Reading Material
Read this as requested, we just make the links available here.
- The best source is the book Geometric Algebra for Computer Science, but that will only be available at the end of the month.
- We have a triple of introductory articles on Geometric Algebra published in IEEE Computer Graphics and Applications 2002 and 2003. This material is made available for limited disctribution, under the usual restriction imposed by IEEE copyright conditions (i.e. keep it to yourself).
The papers CGA1 and CGA2
roughly cover the basics in Part I of the book.
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The conformal model and its use in Part II of the book may be found summarized in the
paper
An Algebraic Foundation for Object-Oriented Euclidean Geometry, L.Dorst and D.Fontijne, proceedings of ITM/ga2003 RIMS Kokyuroku, vol 1378, Kyoto, Japan, pp.138-153, 2004.
-
About the implementation, roughly covering Part III of the book:
CGA3 (same restrictions as above) in combination with
Gaigen2: A geometric algebra Implementation Generator, by Daniel Fontijne (Generative Programming and Component Engineering, 5th International Conference on Generative Programming and Computer Engineering, pp 141-150, 2006).
Assignments
If you choose to take the Geometric Algebra part of the course as your subject, any option on doing a presentation or report can be discussed. It is of course most interesting if you can select something related to your own work. General suggestions are:
- Comparison of methods for rotation (matrices, quaternions, versors), in terms of speed, convenience, etc. A hybrid method may result.
- New operations in Euclidean geometry: expand the techniques sketched in the RIMS paper quoted above.
- Non-Euclidean geometry: spherical geometry is also included in the conformal model. It would be good to embed some of the standard techniques as a straighforward but useful exercise.
- Non-Euclidean geometry: hyperbolic geometry is also included in the conformal model. It would be good to embed some of the standard techniques as a straighforward but useful exercise.
- Interpolating Euclidean motions: even in GA, Euclidean motions cannot be covariantly interpolated due to a fundamental mathematical impossibility on the metrics for the Euclidean group. Help me understand this issue.
For each of those, Leo Dorst can point you to specifically helpful literature.