Clifford Algebra - Generalisation of 2d/3d Formulas

[Roquen]

Or skip on the paper and go to the implementation: http://sourceforge.net/projects/g25/

[kram1032]

that works too, of course. I figured he'd want to do it himself.

[cKleinhuis]

right, i am more interested in the implementation right now, getting it work and play with the parameter space, although i am having kind of a break right now, i will definately try this in the next weeks!

[kram1032]

there is also this: http://versor.mat.ucsb.edu/

[hermann]

Hallo kram1023 thanks for the paper

hermann, for implementing GA, this paper may be of help http://staff.science.uva.nl/~fontijne/phd/fontijne_phd_compressed.pdf

Also thanks to Roquen for the link:

Or skip on the paper and go to the implementation: http://sourceforge.net/projects/g25/

Sorry to  cKleinhuis

right, i am more interested in the implementation right now, getting it work and play with the parameter space, although i am having kind of a break right now, i will definately try this in the next weeks!

I only tried to write down some idears on how to develop a generic package for different geometric algebras. (inspiered by kram1023's thread on geometric algebra).
When you want to get it work fast maybe the link of Roquen is helpful.

I also think about changing the structure of the generic package.
It is easy to define the basic vector in enumeration form for two and three dimensions:
   type Base_Type_2D is (scalar, e1, e2, e12);
   type Base_Type_3D is (scalar, e1, e2, e3, e12, e23, e31, e123);
for Four Dimensions it becomes more difficult:
   type Base_Type_4D is (scalar, e1, e2, e3, e4, e12, e13, e14, e23, e24, etc. )
for higher dimensons it becomes even more difficult.

So I have to rewrite the code in the following form:

Code:

generic
   Dimension: Positive := 3;
package Geometric_Algebra_Generic is

That would allow me to define a geometric algebra of higher dimension very easy.

Code:

package Geometric_Algebra_5D is new Geometric_Algebra_Generic (Dimension => 5);

It would also require that the Basic Vectors of the Geometric Algebra will be generated during the elaboration of the generic package.
That is all not very difficult but very time consuming. May be I have some more time next weekend.

Thanks for the inspiration.
Hermann

[kram1032]

note that one problem with GA is that, if you want to go for a generic, naive implementation, it's really high-dimensional. For instance, the R⁵-GA Cl_5,0 already has 32 different elements, even if most of the time, most elements would be 0. You gotta write sparse code.

[Roquen]

Exactly.  That's why I'd be tempted to suggest a CAS library instead and make code from the final result.

[cKleinhuis]

2d and 3d variants will be enough i think formulating complex and mandelbulb in 2d/3d is what i aim for

[Roquen]

I've been meaning to play around with generalized complex and see what that looks like.  I'd expect like figures 1-4...figures 5 and 6 not though.  Humm.....  (I only quickly skimmed the paper).

Tossed together a generalized complex and maybe it's completely equivalent.  These should be all re-expressible as the standard m-set with a change of domain.  Probably just completing the square...humm....

[Roquen]

Oh and here's a symbolic version in python https://github.com/brombo/GA  (haven't looked at it)