From: "Martin Baker"
Subject: affine transformations using Clifford/Geometric algebra
Date: 09 July 2003 19:08
Could anyone suggest how I might model affine transformations using
Clifford/Geometric algebra
My background is more of a programmer than a mathematician and what I am
trying to do is to model solid objects with simple Newtonian mechanics. What
I would like to do is model all the quantities using multivectors so that
the linear and rotational properties can all be handled in one equation, I
don't want to do anything advanced like spacetime or quantum theory, just
simple mechanics of solid objects. As I say I'm not a mathematician but I
would like to understand the principles behind this.
From reading David Hestenes book, and other material on the web, I gather
that transforms are done using the following formula:
P2=m * P1 * m^1
Where:
P1 = original position of point
P2 = resulting position of point after transform
m= multivector
m^1 = inverse of m
So using 3D multivectors I multiplied out all the terms and found that the
result is the sum of two vectors and a bivector. The first vector is
equivalent to using a quaternion made up of the scalar and the bivector and
the second vector is equivalent to using a quaternion made up of the
trivector and the vector.
My working for this is here:
https://www.euclideanspace.com/maths/algebra/clifford/transforms/
My question is that I want to be able to represent the position of solid (6
degree of freedom) objects, rotation + translation, but instead of the sum
of a rotational and a linear component I have ended up with the sum of two
rotational components?
So how do I model affine transformations? What I have done so far does not
seem to have any advantage over quaternions? What use are two quaternions
added together?
I gather that the same trick can be used as with matrices, i.e. add a dummy
4th dimension to represent the translation component (I have also seen it
suggested that as well as the 4th dimension to represent the origin point, a
5th dimension can be added to represent a point at infinity?).
However this is a problem, firstly I don't like the idea of using 16 (or 32)
scalar values to represent a 6 degree of freedom object. This seems like a
big overhead in memory usage and in normalising all these dummy variables.
Also I can't find any information on the arithmetic of 4D multivectors and
how to represent rotation, translation and scaling using them.
Can anyone suggest how I could take this forward?
Thanks for reading this far!
Martin
From: "Patrick Reany"
Subject: Re: affine transformations using Clifford/Geometric algebra
Date: 09 July 2003 22:58
I belive that you could look at some other of Hestenes's papers found at:
http://modelingnts.la.asu.edu/html/ComputationalGeometry.html
and
http://modelingnts.la.asu.edu/html/UAFCG.html
Patrick
From: "Martin Baker"
Subject: Re: affine transformations using Clifford/Geometric algebra
Date: 10 July 2003 08:10
Thanks Patrick,
There are a lot of interesting documents here and I may have missed
something but I can't find any ideas about how to model affine
transformations.
Have you come across a document which covers this?
Where I am stuck is, do I have to use 3,4 or 5 D vectors? Which part of the
multvector holds the translation? And what goes in the other parts of the
multivector?
Martin
From: "Patrick Reany"
Subject: Re: affine transformations using Clifford/Geometric algebra
Date: 10 July 2003 20:29
Unfortunately, I don't have much time to answer your technical
questions, but I think you couldn't do any better than to start with
the paper
http://modelingnts.la.asu.edu/pdf/CompGeomch1.pdf
and take particular notice of the section "Linearizing the Euclidean
group" on page 20. As I recall, the vectors are in 4 dimensions.
Patrick
From: "M.T."
Subject: Re: affine transformations using Clifford/Geometric algebra
Date: 21 July 2003 13:27
don't know if this helps, but if one constructs the clifford algebra
on a quadratic space V(Q), then the unit ball of the resultant even
subalgebra rotates V(Q) in the form you gave. so unit complex numbers
rotate R^2, and unit quaternions rotate R^3.
M.T.
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