How can I create a ring torus using geometric algebra?

[Iason Solomos]

This question was also posted on quora and mathxchange, however I did not find any help there yet. I would like to ask the following:

I'm not sure how to use the notions of the vector outer product or the multivector operations in order to achieve the ring torus shape.
The ring torus is something like this (made in MATLAB):

[Wesley Smith]

You can use 2 circles from the 5D conformal model an axis using a rotor.  Have you read GA For Computer Science?  I'd suggested looking there.  You'll find all of the math to work this our by looking at the chapters on conformal GA.

Have a look at Versor for how you would program such a thing.

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[Iason Solomos]

Thank you for the answer. I am not looking to code it actually, it is more of a concept question, as I am quite new to GA. Really my question is how. How should I make 2 circles in 3 dimensions, using GA, so that I can create a ring torus. Since I am going to use trivectors, does it mean I have to know 8 variables (scalar, 3 basis vectors, 3 basis bivectors and trivector)?

[Fred Lunnon]

This is probably more general than you had in mind,
but might give you some useful ideas ...

"Lie-sphere algebra in 3-space and a fresh look at the cyclides of Dupin"

https://www.dropbox.com/s/f73l1gguynuqiyz/dupin.pdf?dl=0

Fred Lunnon

[Iason Solomos]

Thank you. It is unfortunately quite general.

[Lanco]

I think we need to know about your background in geometric algebra (e.g. books etc.).
Also how you imagine the torus represented:
1. As a parametrized surface ?
or 2. As an element or a construction in a geometric algebra
In this case which sort of  geometric algebra
2.1 based on a 3D euclidean vector space
2.2 based on a 4D space (homogeneous GA)
2.3 based on a 5D space (conformal GA)
3. Should it be drawn?
Lancoz

[Fred Lunnon]

Or indeed the natural coordinate system for such objects,

2.4 based on a 6D space (Lie-sphere GA)

WFL

[Iason Solomos]

I have read only a little from various sources like Hestenes and Suter. I think what I want is 3 dimensional Euclidean space

[Lanco]

To work with circles and spheres as GA-elements conformal geometric algebra or higher are adequate.
Tori are more complicated. Topologically, as you know, a torus is a closed surface defined as the product of two circles: S1 x S1.

The Handbook of Geometric Computing by Eduardo Bayro Corrochano  at
 http://newplans.net/RDB/Handbook%20of%20Geometric%20Computing%20-%20Eduardo%20Bayro%20Corrochano.pdf has
has in chts. 22.11-12 vector calculations for the spherical torus and the general torus.

 Another link interesting link could be about the Clifford torus in 4D:
 https://en.wikipedia.org/wiki/Clifford_torus
 
 Lanco

[Robert Easter]

Hi, if you look at this paper on "G8,2 Geometric Algebra, DCGA" that I wrote recently

http://vixra.org/abs/1508.0086

it provides a way to represent a torus which can be positioned generally using rotors, translators, and a dilator. It also allows to intersect the torus with spheres, planes, lines, and circles (CGA entities). You can use this DCGA algebra in SymPy (the setup is not too hard), and use it to generate the fomula for the toroid after doing operations on it. Then, it is possible to graph the resulting torus equation using software like MayaVi. I had some fun playing around with this myself. I'm curious to get some peer-review feedback on my DCGA paper and this is the first posting anywhere about it other than the paper itself on viXra.org.
- Robert Easter

[Lanco]

It seems that your fine work answers Ianos question - and much more.
A conformal geometric algebra Cl(8,2) called DGCA is used to give an entity expressing the toroid dually and tightly connected to equation for the toroid with help of extraction operators.

A remark seems adequate, that the geometric outer- and inner-product null spaces are denoted GOPNS and GIPNS, respectively.
The notation from Perwass seem to be used with dot product as bi-contraction.
It will be nice to have T_D on p. 11 expressed instead or also with e_inf's  and e_o's and the t-entities on p. 12.
This will make it easier to see how the extraction operators work.

lanco