CMPS 161 Final Project Winter 2007

Name: Chirag Dave

Email: cdave at ucsc dot edu


Physics using Geometric Algebra


Project Description

Implement basic physics engine using Geometric Algebra. Lots of Collisions, lots of speed, lots of accuracy. The Geometric Algebra library will be implemented using Gaigen.


Goals

-Show the world that Geometric Algebra is good for you.

-Calculate intersection between ball and circle, accurately.


Screen Shots

With just four points, using the geometric algebras outer product:

sphere = a^b^c^d in conformal geometric algebra you get a sphere.

With 2 and 3 points, you can get circles, and point pairs.

circle = a^b^c

point pair = a^b

A circle wedged with the conformal geometric algebras infinity gets you a plane. And point pairs wedged with it, will get you a line.


Using the dual operation, you can get the compliment of any multi-vector. In this picture the red sphere is the dual of the yellow sphere. If you do the following dot product: dual(R)*Y,

you will get the intersection, which is the circle.







In this picture you have an intersection between two spheres, and the plane.


With an optimized GA library, these intersection tests can be done much quicker, and much more accurately than using standard methods.


















Videos


[ Put mind blowing videos here ]


Timeline

This project can be broken up into 4 steps:

  1. Learning Geometric Algebra's details, figuring out how to apply it to physics.

  2. Setting up Gaigen properly

  3. Writing physics engine.

  4. Documentation

I would say that 1) will probably be going on through out the entirety of the project. Where as 2) will be trivial. I just need to learn how to use it. And make sure I use the right type of Algebra for my solution.

And 3) Will be going on from when 2) is complete and ending when the project is due.

I am hoping 4) will go on throughout the project...but I think the world knows that will never happen.



References

“Geometric Algebra for Physicists” - Doran, Lasenby

“Primer on Geometric Algebra” - Hestenes

“Geometric Algebra and its Application to Computer Graphics” - Hildenbrand

“Physical Modeling using Geometric Algebra” - Cameron

“Computer Graphics Using Conformal Geometric Algebra” - Wareham

“Imaginary Numbers are not Real” - Doran, Lasenby

“Recent Applications of Conformal Geometric Algebra” - Lasenby

“Gaigen: a Geometric Algebra Implementation Generator” - Fontijne