GA and Space Navigation: Which model to use?

[Joel C. Salomon]

Neighbors,

Several years ago, I asked here about resources to learn Geometric
Algebra from, with a focus on navigation in space (the soft-landing
problem, e.g.). Since then, any actual work on my thesis has fallen by
the wayside, though I’ve continued to scan various books & articles on
the subject.

I’m trying to give myself another push to actually get back into the
math: My plan is to learn enough GA to follow W. T. Thomson’s
“Introduction to Space Dynamics”
(or any text on the topic, but that’s the one I own) keeping my notes
in GA form.

Thing is, I’ve perused enough to know about the different models:
“plain” 3D, relativistic variants, Fontijne et al.’s conformal model for
computer graphics –– which do I use for this?

(Can the conformal model ℝ⁴⋅¹ even be made compatible with relativistic
models ℝ³⋅¹ or ℝ¹⋅³ without introducing a second negative dimension,
ℝ⁴⋅², and losing a bunch of nice mathematical properties of Minkowski
space? Or can the time dimension do double duty, since motions &
rotations are anyhow restricted?)

Whatever model I choose, I’ll have a tough enough time with the math
that’s required. Maybe down the line I’ll find it useful to pick up a
more complex model, but for now I want to focus my efforts on the
simplest model which will suffice –– but again: which variant do I need
here?

Texts: I own Hestenes’s “New Foundations for Classical Mechanics” and
Fontijne et al.’s “Geometric Algebra for Computer Science”; and I still
have the 2009 draft of Macdonald’s “Linear and Geometric Algebra”. Last
time I posed this question, I had Doran & Lasenby’s “Geometric Algebra
for Physicists” recommended to me, but whether as an e-book or used
dead-tree, the prices online are pretty stiff: is this still the best
option for me, though? (I’m trying to get my local public library to
get this in as an interlibrary loan; failing that I’ll take a trip to my
alma mater and see what alumni borrowing privileges I still have.)

—Joel C. Salomon

[Fred Lunnon]

What "nice mathematical properties of Minkowski space" are you concerned
about sacrificing by adopting Cl(4, 2) ? For (non-physics) 3-space
geometrical
computations, I nowadays rarely employ anything else!

Fred Lunnon

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[Joel C. Salomon]

On 2017-05-05 7:14 AM, Fred Lunnon wrote:
> What "nice mathematical properties of Minkowski space" are you concerned
> about sacrificing by adopting Cl(4, 2) ? For (non-physics) 3-space
> geometrical computations, I nowadays rarely employ anything else!

According to Dorst, Fontijne, & Mann, “Geometric Algebra for Computer
Science”, “only in Euclidean and Minkowski metrics rotors can be written
as the exponentials of bivectors.” (See §7.4.3 for more details.)

I may well be reading more into this limitation than actually matters;
I’ve been scanning ahead of what I actually understand, in hopes of
figuring out what it is I really need to learn. If all the limitation
means is that ›some‹ rotors in Cl(4, 2) cannot be written as
exponentials of bivectors, but that exponentials of bivectors still do
generate rotors, this matters less. (Dare I hope that most rotors of
practical interest can still be generated that way?)

(What, may I ask, are you using Cl(4,2) for if not for physics? Is there
another application of a second negative-signature dimension?)

—Joel C. Salomon

[Manfred]

On Friday, 5 May 2017 08:19:31 UTC-7, Joel C. Salomon wrote:

According to Dorst, Fontijne, & Mann, “Geometric Algebra for Computer
Science”, “only in Euclidean and Minkowski metrics rotors can be written
as the exponentials of bivectors.” (See §7.4.3 for more details.)

I may well be reading more into this limitation than actually matters;

The Euclidean and Minkowski metrics are not special.  As far as I know, every rotor in the connected component can always be written as the exponential of a bivector.  Going to higher dimensions does not lose any rotors, provided the representation you choose does not lose the metric.  (An example of the latter might be the homogeneous model, where the rotors that produce displacements do not preserve the original metric).  In the Minkowski metric, I suspect that time-reversing rotors might not be expressible via exponentiation.
 

I’ve been scanning ahead of what I actually understand, in hopes of
figuring out what it is I really need to learn.  If all the limitation
means is that ›some‹ rotors in Cl(4, 2) cannot be written as
exponentials of bivectors, but that exponentials of bivectors still do
generate rotors, this matters less. (Dare I hope that most rotors of
practical interest can still be generated that way?)

I expect that "most rotors of practical interest", if defined as those that preserve time direction, can be generated by exponentiation of bivectors. 

Note that a benefit of Cl(4,2) is that its rotors express the full Poincaré group as (a quotient of) a subgroup, and is probably the smallest GA that does so.  That is, it models the Poincaré group of Minkowski homogeneous space, whereas Cl(3,1) only expresses the Lorentz group of the Minkowski vector space.  But take this with a pinch of salt: I'm using general reasoning, and I might have some finer points mixed.  A limitation of Cl(4,2) is that it cannot express general linear transformations of space.  Outermorphisms are used for this, and they go outside the algebra, but then, I've seen no published results for any GA to achieve GL transforms without outermorphisms.

(What, may I ask, are you using Cl(4,2) for if not for physics? Is there
another application of a second negative-signature dimension?)

It might help to be clear on what manipulations you want to achieve, such as:

• All transformations of a vector space that preserve the scalar product (Cl(p,q) needed)
• All transformations of a homogeneous space that preserve the distance function (Cl(p+1,q+1) needed, though all conformal transformations come for free)
• If you want to go outside the conformal group, clearly more will be needed.

Manfred

[Eskki Paramasivan]

Dear Joel C Salomon

your wonderful probe in space dynamics is very much mind-boggling.

May I expect an  expertise in the form of treaty on ASD/CFT model that too

both founded on self consistent ten dimensional Super String Theory and on also the classical  complex conformal mapping with "Zukovsky's Function.

with regards

E.Paramasivan

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