# Talk:Conformal geometric algebra

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## Collection of references

Maybe we can collect up a bunch of refs to start with?Selfstudier (talk) 18:52, 6 February 2012 (UTC)

## Subject Intro

JHeald, the lead in refers refers to computational geometry which is (according to Wikipedia) a branch of computer science not mathematics (follow the link).

As you rightly point out the field is not restricted to computational geometry which is why I deleted the lead in the first place since you have covered applications in paragraph 4 (I don't know about "good practice" but it seems to me that anywhere in the into will do, as is the case with the GA article).

If you insist on keeping the lead, it will need amending somehow to reflect all of the applications or else delete the reference to mathematics, or something else; it is not sensible the way it was so I have deleted it again.Selfstudier (talk) 19:38, 7 February 2012 (UTC)

I'd say its classification is the same as that of geometric algebra. It is also no more computational geometry than linear algebra is. Of course, it does have computational applications, but that is its applications, not the subject as a whole. I'd suggest leaving "computational" out of the lead altogether. — Quondum 20:19, 7 February 2012 (UTC)
WP:CONTEXTLINK: "In an article about a technical or jargon term, the opening sentence or paragraph should normally contain a link to the field of study that the term comes from."
It then gives an example which starts "In heraldry..." -- this kind of opening starting with "In..." is very much the generally found way of achieving this.
As I wrote in my edit summary, there is solid thinking behind the recommendation. People can come to a Wikipedia article in all sorts of ways (including "Random article"), and it makes sense to establish the frame of context for the article right from the start.
I hope we can agree that "mathematics" is probably the most appropriate general subject area. But according to the Maths project, it is useful to also give a more specific area of maths to more precisely set the context.
My suggestion was "computational geometry", because the practical usability has been the focus of most of the material published on the subject -- eg Dorst et al (2007), Perwass (2009). Quondum is quite right, that you don't have to use it to compute with. But that is the field whose people have come forward and run with it, and which, I submit, it's been most associated with. Yes, it spills over beyond that; but I suggest that if we're trying to give people a one or two word fix on where this material is coming from, & where it seems to have a particular value, then "computational geometry" does that quite well. It will become apparent soon enough that CGA is pretty abstract, and not just reduced to bits and bytes and floating point. Jheald (talk) 22:47, 7 February 2012 (UTC)
I would however consider "projective geometry" as a possible alternative; though there are problems there too. Jheald (talk) 23:09, 7 February 2012 (UTC)
If someone were to ask me what the field of study was, what I would really want to say is "geometric algebras" (uncapitalized), this term being well understood by all those applied folk in computational geometry, robotics and graphics, etc and I think by mathematicians too (even those of a Clifford algebra persuasion),certainly by physicists.

And then that CGA is an example, like "basic" GA, of such a geometric algebra (I don't want to open up that whole can of worms again but I sometimes wish there was a page called "geometric algebras" and a summary of those, maybe I will do one:-).

In some ways, the "first" CGA (going by the definition given) is R^(3,1), which presumably would be the "conformal geometry" (another candidate for field of study) of R^2. Hmmm.

I still have the same objection, computational geometry is too narrow (and isn't mathematics); what about just "classical geometries"?Selfstudier (talk) 23:51, 7 February 2012 (UTC)

There is "geometric algebra (disambiguation)". ((Now there's a really unhelpful hint as to what GA is!)).
The rest of your cmt I'll have to get back to. But I disagree that most people in those disciplines would know what GA (or even ga) is, or would even have heard of it. Jheald (talk) 00:04, 8 February 2012 (UTC)
I agree that most would not know what GA is, but all of a certain age know what a "geometric algebra" is (they have to know to do their work)Selfstudier (talk) 00:15, 8 February 2012 (UTC)

Jheald, could you exand on what you mean by "I would however consider "projective geometry" as a possible alternative; though there are problems there too"? (Though I too have reservations).

I agree with the format that starts "In ...". If we were to use "computational geometry", we'd have to at least reword it to something like "With application in the field of computational geometry ...", which doesn't quite work. It seems to me that the definition of GA includes anything that fits the math, including projective geometries. As such, CGA, is simply a part of GA. And to put "In geometric algebra ..." is too specific. I'd suggest "In mathematics ..." until we find a suitable narrower field. Or, if you really want to give people a better fix on the field, how about "In mathematics, with application in computational geometry ..." (though I think that is getting unnecessarily verbose)? — Quondum 07:34, 8 February 2012 (UTC)

How about Geometry and Computing? Selfstudier (talk) 11:26, 8 February 2012 (UTC)

Both of these seem incorrect. CGA does not fall into the field of geometry. Nor does it fall into the field computing. At least IMO. — Quondum 12:12, 8 February 2012 (UTC)
You're probably right, seems we are hamstrung by "good practice", requiring us to fit square pegs into round holes...

at least it's not wrong now, just clunky.Selfstudier (talk) 12:44, 8 February 2012 (UTC)

OK, going back to the intro again, a slightly different issue in the first bit. We say that a "geometric algebra" (linked) results and then that a "geometric algebra" (unlinked) is constructed over....?Selfstudier (talk) 13:01, 9 February 2012 (UTC)

The first sentence should read something like

In mathematics, with application in computational geometry, Conformal geometric algebra (CGA) is geometric algebra (Clifford algebra) established over a conformal space that results from n-dimensional Euclidean (or pseudo-Euclidean) space ℝp,q being projectively mapped into ℝp+1,q+1 in a particular way

Selfstudier (talk) 13:27, 9 February 2012 (UTC)

I really don't see a problem. You have to remember, the lead is meant to be pitched at people who are wildly unfamiliar with the concepts of the article. It's valuable to be as concrete as possible, to make clear exactly what the GA is constructed over. What may seem like redundancy to you, because you are very familiar with the idea of CA and GA, I believe is helpful in making things explicitly concrete to somebody starting in on this article blind.
Can I suggest that rather than critiquing every word of the lead, what is really needed now is to build up a bit more of the body of the article? Jheald (talk) 14:49, 9 February 2012 (UTC)
I wasn't aware that I was in any way preventing you or anyone else from building up the body of the article, please feel free to do so.
I was simply pointing out (politely) that the first sentence could do with a rejig; now I will be less polite and stipulate that it is wrong (as in incorrect) Selfstudier (talk) 15:13, 9 February 2012 (UTC)
It isn't wrong. Structurally the sentence is much the same as saying that "X is the bus that results if a traffic flow is selected at random and then a bus is chosen from that traffic flow using PQRS procedure". What is the problem? Jheald (talk) 15:24, 9 February 2012 (UTC)
It says a geometric algebra "results" (from....) and then says that you need to construct it ie the sentence says you need to construct a geometric algebra on a geometric algebra which makes no sense, the algebra is constructed on a space.
What is wrong with my suggested rewording? Or something similar? I could have simply made an edit but thought instead to discuss it first.Selfstudier (talk) 15:32, 9 February 2012 (UTC)
It says it is the algebra which results from a particular procedure, then describes the procedure, including the point at which the algebra is constructed. I don't see why you want to suppress some of this, and make the intro less immediate, less concrete, and less informative. Jheald (talk) 15:40, 9 February 2012 (UTC)
What is less immediate, less concrete and less informative about my proposed edit? IMHO, it is more informative not to mention shorter.
I am coming round to the view that you are objecting to these edits simply because you feel some kind of ownership of the article notwithstanding that it is now in mainspace (community space) rather than your sandbox.
Would you feel better if I write up an alternative intro instead and then let you (and others) edit it?Selfstudier (talk) 15:51, 9 February 2012 (UTC)
Well, let's see what other people think first, and whether they agree with you; or whether they find the lead clearer as it is currently. Jheald (talk) 15:59, 9 February 2012 (UTC)
I find the current intro understandible. The phrase "over a conformal space" immediately loses me, as I'm sure it will lose the majority of the readers. That concept should be left for the body of the article. The lead will need a little bit of polishing still – things such as "in a particular way" should be removed. There is also too much detail further on in the lead. I suggest not putting too much energy into getting it just right yet; it is already pretty good. It is going to be the prime target of every editor that reads it still, so your carefully crafted wording might not survive long. — Quondum 18:06, 9 February 2012 (UTC)
Well, a lead should summarise the whole of the articles. Many summarisations of WP (including the official DVD project) only reproduce the lead, so it should aim to be reasonably comprehensive (or as comprehensive as can fit into four paragraphs with as few prerequisities as possible, anyway). I still prefer what was written before. But final polishing of the lead is probably the final thing that should be done on the article, not the first, so I'll leave it for the time being, until more of the article is written. Jheald (talk) 13:35, 10 February 2012 (UTC)

## Meaning of Conformal

I cannot help but feel that the word "Conformal" in the name (the C in CGA) ought to be explained somehow, reference is made two sentences later to conformal mapping (linked), I would assume that those people getting lost at "over a conformal space" are going to be equally lost at "conformal mapping" (particularly when they follow the link and find themselves reading about "complex analysis" and "Riemannian geometry"). Conformal space presently redirects to conformal geometry which at least attempts to cover Mobius geometry.Selfstudier (talk) 18:49, 9 February 2012 (UTC)

Yes, I noticed that too, and considered it. With that I felt a reader would be able to gloss over the not-understood adjective and just read "mapping" for "conformal mapping". But I'd prefer to remove "conformal" there too, just felt that that was to be dealt with later as less significant. Somehow "conformal space" is not as easy to gloss over: it seems to carry some import. Besides, I know what a conformal mapping is. I can even define it in a few words. I haven't the faintest idea of what a conformal space is. — Quondum 19:05, 9 February 2012 (UTC)
For quick consumption, this is OK [1]Selfstudier (talk) 19:23, 9 February 2012 (UTC)
I assume then it is the word itself that is the problem, after all we have metric, vector and topological spaces mentioned all over the place:-)Selfstudier (talk) 19:33, 9 February 2012 (UTC)
Essentially, yes. I guess it might not be a great idea to use a potentially unfamiliar word when defining something. It makes it easier for the reader. Besides, I'm thinking of someone a bit less familiar (though not much) than I am with the subject matter. Incidentally, the reference you gave just above (with its reference to Cycles of Time by Penrose) makes me think of Penrose's conformal maps, which are quite different in character from the conformal mappings of CGA. In that sense, I'd say that the representation space of CGA is not a conformal space. Only angles about the origin are significant in CGA – no other feature has any significance. The "C" in CGA seems to me to relate to what is being represented by the transformations within representation space (the conformal/Möbius geometry of the base space), but is not applicable to the representation space itself. I may be off-beam, of course. — Quondum 19:56, 9 February 2012 (UTC)
At the risk of causing more confusion,to quote Perwass, "...when people usually mention the conformal Geometric algebra they actually mean the Geometric algebra over a homogeneous conformal space. We will not break with this tradition here and simply talk of conformal space." Selfstudier (talk) 20:08, 9 February 2012 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── No, that makes (a bit) more sense to me. Especially the "homogenous" part, by which I read "projective". Anyhow, I think my preference would be to avoid terms that make a mouthful; in the lead one can trade preciseness for quick understandibility. The purpose of the lead is to let the reader form a fairly quick impression to decide whether to read further. If you feel understanding "projective", "conformal" and "homogeneous" can be made prerequistes for tghe first paragraph, fine, I won't object – I am only hoping to provide what I hope is a reader-oriented perspective. To me, the article body is more important at this stage. — Quondum 20:46, 9 February 2012 (UTC)

OK, in my case, I will be happy to make a contribution to the main body after I see how the lead finally works out, hopefully by then I will have a much clearer idea of what is and isn't permissible/desirable/good practice etc.In the meantime I will try and compose something up for a History section, shouldn't get into too much trouble with that...:-)Selfstudier (talk) 00:04, 10 February 2012 (UTC)
I've been considering this issue; maybe you can help me on this. In trying to understand what is meant by a "conformal space". It isn't defined by my Collins math dictionary, and the Wikilink just goes to a definition of conformal transformations and geometry.
• In CGA, the use seems to be as not a formal object at all, merely a shorthand (including Perwass's use) to mean a space in which a certain class of transformations (rotations about the origin) maps to conformal mappings within the base space. Note that the mapping ℝp,q → ℝp+1,q+1 itself is not a conformal mapping, though a quotient space of the null vector space of ℝp+1,q+1 could maybe fit the concept of the next points.
• Penrose refers to conformal diagrams, mappings and representations. When he refers to a "conformal space-time", he seems to be referring to a space of the same dimensionality and metric as the base space, and is related to the base space though a specific conformal mapping; in short, a conformal representation.
• If the "quick consumption link" that you gave is to be taken seriously when it says "With conformal space we are concerned with angles rather than distances. We therefore don't need a complete metric which tells us what the distance between any points are, we only need to define enough to allow angles to be defined.", then conformal space is a formal object: essentially one in which distances are "forgotten", without losing information about angles (locally). I have yet to see this class defined formally.
To me, this is significant if a phrase such as "conformal space" is to be used, even in the body, notwithstanding its sporadic use in the literature. I've run into problems with informal shorthand in maths before, and find it problematic in encyclopedia articles. I'd appreciate some pointers here. — Quondum 06:56, 10 February 2012 (UTC)
I will try and write the following up a bit more clearly a bit later but if u look at the actual mapping for a point, then the n_0 is the projective/homogeneous bit and the quadratic part is what guarantees isotropy (pp = PP = P dot P = 0, null).
I don't know if you were following the history discussion but this mapping can be rewritten (per Hest/Sob 1984) as -(x-e_+)n_inf(x-e_+) ie your typical sandwiching.
It is actually the "conformal split" R^(n+1,1) = R^n + R^(1,1) (not sure what the sign between should be, usage varies)
This can also be explained in a (more abstract) group theoretic way, if u fancy it, try to work your way through Hest/Sob 1984 making use of the above info (start at p802 and work backwards)
Anyway, you are right that "conformal" alone tends to give the wrong idea (it is a convenient usage), OTOH that is precisely why I think it ought somehow to be dealt with quite early on so the idea does not stick.
Selfstudier (talk) 11:23, 10 February 2012 (UTC)
The more I look at it, the more I am convinced that the "right" page to link to is the conformal geometry page which does at least contain the relevant info at various points; maybe we could just link the word "conformal" in "conformal geometric algebra" to that page and somewhere later on explain that we are talking about Mobius geometry/flat structures.Selfstudier (talk) 12:17, 10 February 2012 (UTC)
F Morley described conformal space as "the space of the geometry of inversion"Selfstudier (talk) 12:39, 10 February 2012 (UTC)
You may find a discussion of confusion surrounding Penrose/this subject/Clifford algebras etc in Geometry and Shape of Minkowski's Space Conformal Infinity. To me this is YA (yet another) example of how to take something straightforward and unnecessarily complicate it, you might find it illuminating howeverSelfstudier (talk) 12:55, 10 February 2012 (UTC)
I remember seeing the sandwiching form for the mapping into the CGA somewhere (have lost it now), but couldn't see what it added to understanding. Its interpretation is a little obscure – it looks like reflecting the point at infinity along a vector determined by the base space vector. I find the other expression more intuitive. I didn't follow the history discussion closely, and missed that (history isn't a huge interest of mine). The page number seems a bit odd: Hest/Sob 1984 only goes to p.314, so p.802?
The idea of linking [[Conformal geometry|conformal]] geometric algebra seems to have merit. The applicability can be explained in an introduction section. The applicability of CGA in representing Möbius/conformal geometry is very clear.
I'll look at arxiv 1107.0933, though a glance at the abstract suggests it is dealing primarily with the point at infinity in a conformal setting, and that the paper may be a little abstract for me, but we'll see. — Quondum 13:12, 10 February 2012 (UTC)
Yes, sorry, 302 not 802; the sandwiching and the mapping (for general points) are the same so EVERY operation is a combo of reflections ----> conformal. I tend myself to concentrate on (4,1) because that is used a lot and I am not aware of other meaningful applications at this point (which is not to say that there might not be later on).Selfstudier (talk) 13:42, 10 February 2012 (UTC)
Okay, found it. That's not the one I remember; it was no doubt a later quote of this instance. The idea of everything being a versor operation is nice, but I don't think that's quite the case: there are the various products within the CGA, and the normalization. But expressing everything in the algebra in terms of set of multiplicative and sandwiching operations would be a great plus. And this does not necessarily detract from your idea of implied conformality. — Quondum 14:19, 10 February 2012 (UTC)
You can go down the page to JHealds' Transformations section to understand it moreSelfstudier (talk) 14:35, 10 February 2012 (UTC)

## Möbius geometry and CGA History

The projective model of Möbius geometry has been known for long, I have a couple open references for it somewhere and I will dig them out; this is essentially CGA, Bobenko even makes use of the null basis trick in a 2008 paper.Selfstudier (talk) 19:54, 7 February 2012 (UTC)

Here is a good recent reference for the CGA ProjMG correspondence [2] It's a bit unclear to me why ProjMG has been coopted by the (discrete) differential geometry community....Selfstudier (talk) 15:19, 16 February 2012 (UTC)

Technically we'd need references that discuss the correspondence. We can't go about doing that, only report on someone else doing it. — Quondum 20:21, 7 February 2012 (UTC)
Yes I take your point, I'll still supply the refs just for interest sake, though. I'm not sure, I think when they go on about this Wachter fellow, I think they must be referring to the subsequent circle and sphere geometries of Laguerre, Mobius and Lie, can't think what else it might be.Selfstudier (talk) 20:39, 7 February 2012 (UTC)
Mind you the conformal geometry page has quite a lot of it except that it has mixed it altogether with the 2 dimensional stuff in a very confusing way and tried to treat both curved and flat as one (another of those wrong headed generalizations).Selfstudier (talk) 20:47, 7 February 2012 (UTC)
Perwass (2009), p.4 says of CGA:

Incidentally, Pierre Anglès had already developed this representation of the conformal model independently in the 1980s [6, 7, 8] but, apparently, it was not registered by the engineering community. His work on conformal space can also be found in [9].

where the refs are
6. Anglès, P.: Construction de revêtements du groupe conforme d’un espace vectoriel muni d’une métrique de type (p, q). Ann. l’I.H.P., Section A 33(1), 33–51 (1980)
7. Anglès, P.: Géométrie spinorielle conforme orthogonale triviale et groupes de spinorialité conformes. In: Report HTKK Math., vol. A, pp. 1–36. Helsinki University of Technology (1982)
8. Anglès, P.: Algèbres de Clifford c+(r, s) des espaces quadratiques pseudo-Euclidiens standards er, s et structures correspondantes sur les espaces de spineurs associés; plongements naturels des quadriques projectives associées. In: J. Chisholm, A. Common (eds.) Clifford algebras and their applications in mathematical physics, C, vol. 123, pp. 79–91. Reidel, Dordrecht (1986)
9. Anglès, P.: Conformal Groups in Geometry and Spin Structures. Progress in Mathematical Physics. Birkhäuser (2006)
I don't know whether any of those mention Möbius geometry, but it shows that at least one author is recognised as having published this construction before Hestenes et al
I don't see what's wrong with presenting Möbius geometry and appropriate references for readers to make the comparison, if it's a close relative. Why not knock up a draft of what you have in mind, so we can see what we think? Jheald (talk) 23:04, 7 February 2012 (UTC)
I have the last one, I will take a look although I seem to remember it being a bit hard going (maybe why the engineers didn't pick it up? Not sure I really approve of that comment from Perwass). Fact is the mathematics crowd dropped the ball on all this and as you say, it had to be picked up by physicists and then by CS folk (in order to get an implementation).Selfstudier (talk) 00:11, 8 February 2012 (UTC)
I can't lay hands on the two French publications prior to 1984 when Hest/Sob was published; going by the titles and looking at Angles 2008, I have my doubts about whether the construction precisely matches the Hest/Sob construction, you could equally say that everyone who wrote about Mobius geometry also provided the essential construction, thing is none of them ran with it or properly established and calculated with a GA on the space. Also it is claimed that Hest/Sob manuscript was actually completed in 1975 (they had a lot of publishing problems).
I remember taking that disambiguation thing out when I redid the GA page, then it mentioned algebraic geometry as a possible confusion...heh!Selfstudier (talk) 00:21, 8 February 2012 (UTC)
CGA isn't in CA to GC (Hest/Sob 1984). It first appears in the four Hestenes et al papers at the start of Sommer (2001), which are linked in the resources section of the article. The patent (US 6,853,964) gives a priority date of June 2000. Jheald (talk) 12:18, 8 February 2012 (UTC)
"Except for the last chapter added in 1979, the manuscript was ready for publication by 1976 but did not appear in print until 1984, owing to an unfortunate series of publishing mishaps." [3]. Jheald (talk) 15:26, 8 February 2012 (UTC)
Yes it is, look at the last pages of the last ChapterSelfstudier (talk) 12:48, 8 February 2012 (UTC)
Bottom of p301, start of p302....Selfstudier (talk) 12:54, 8 February 2012 (UTC)
Very interesting. How close is the mapping represented by eqn (3.14) to the equation we have on our page? Very naughty not to cite this to the patent examiner as prior art (though the patent probably isn't worth a lot anyway). Wonder why it took another sixteen years before Hestenes & co suddenly lit the touch paper on this.
It's essentially the same, you can look at [4] for a translation between.... I don't remember where it was exactly, one of H's papers where he 'fessed up to not realizing what they had at the time (physicist + mathematician not equals computational geometry:-) They were intent on the calculus at the time so I guess it just went straight past them. I don't know for sure but I think it wasn't till Li got into it along with Hestenes that they really figured it out :-)
Selfstudier (talk) 13:37, 8 February 2012 (UTC)
I found this on p359 of Dorst GA for CS:
"Even though the conformal model hit computer science only a few years ago, when it was introduced by Hestenes et al. in 1999 [31], we are beginning to ﬁnd that we could have had the pleasure of its use all along. Considerable elements of it are found in much older work,and it appears to have been reinvented several times. We do not know the whole story yet,but markers on the way are Wachter (a student of Gauss), Cox [9] and Forder 1941 [22](who used Grassmann algebra to treat circles and their properties), Angles 1980 [2] (who showed the crucial versor form of the conformal transformations) and Hestenes (who had already presented it in his 1984 book [33] but only later realized its true importance for Euclidean geometry)Selfstudier (talk) 00:11, 12 February 2012 (UTC)
or Alyn Rockwood? He is a computational geometer. Jheald (talk) 14:36, 8 February 2012 (UTC)
Incidentally, the patent specifically references F. L. Wachter (de.wiki)'s introduction of the horosphere, showing that "a certain type of surface in hyperbolic geometry is geometrically equivalent to Euclidean space". So if you wanted to write a "History" section for the end of the article, that would certainly give you cover to talk about Wachter, and to trace subsequent developments. Jheald (talk) 13:23, 8 February 2012 (UTC)
Ah, the horosphere...OK, that makes sense, thanks very much for thatSelfstudier (talk) 13:40, 8 February 2012 (UTC)
Bonola, Non-Euclidean Geometry, (tr. by H.S. Carslaw, Dover, 1955), [5] and foot of [6] best ref I could quickly find for Wachter's contribution....Selfstudier (talk) 15:58, 8 February 2012 (UTC)
Coxeter gives a similar snippet [7]. Hitzer [8], also [9], quotes this along with the snippet you gave.
Our article on horoball doesn't seem particularly enlightening, and should perhaps be re-orientated towards the older, simpler term. (We also have horocycle). Google books returns various more hits from other books on non-Euclidean geometry, eg [10] contains a bit more. According to Encyclopedia Britannica, "[t]his surface, now known as the horosphere, was destined to play a vital role in the development of the subject." [11]. The term also occurs in EB here.
A key thing for any historical background would be to establish whether the isomorphism
C(r,s) ≅ O(r+1,s+1) [Hestenes 2009 [12], eqn. 16]
-- i.e. between the conformal group in Rr,s and the orthogonal group in Rr+1,s+1 -- was well known and appreciated before Hestenes et al (2000); and whether concrete constructions or specific maps had been discussed to demonstrate it. Jheald (talk) 18:05, 8 February 2012 (UTC)
After some more investigations, I believe so - can you get this paper (front cover is quite explanatory though and they describe the isomorphism as well known which I think is right )[13] This relates to the Hestenes 1991 Design of Linear Algebra and Geometry. So it isn't Li (or Rockwood) then, perhaps Dress/Havel was the spur. Also, Hestenes doffs a cap to Angles and refs his 1980 paper that you showed above together with the comment that Angles studied it but does not make use of the conformal split to simplify the analysis.
I think that what Hestenes has done is to take a bit of his own work and a bit from here and a bit from there and a bit from everywhere and then made something real out of it; I realise some mathematicians might be a little miffed that he doesn't quite play by the rules (he's a physicist so must be forgiven:-) but I for one am happy to give him credit where credit is due.Selfstudier (talk) 18:58, 8 February 2012 (UTC)
Havel also did a paper that I can't find anywhere "Geometric Algebra and Mobius Sphere Geometry as a basis for Euclidean Invariant Theory"Selfstudier (talk) 19:28, 8 February 2012 (UTC)
The citations say it's in Neil L. White (ed, 1995) Invariant Methods in Discrete and Computational Geometry, pp. 245-256. None of the universities in London have it, but there's a copy in the British Library if you really want it. Easiest thing might be to email him at MIT [14] and see if he's still able to send you a copy.
Incidentally, I see that in 1999 Li wrote a paper Some Applications of Clifford Algebra to Geometries -- This was specifically looking at the "symbolic representation of geometric entities, such as points, lines, planes, circles and spheres", and it may be that it was realising how much CGA had to offer in that context specifically, that led Hestenes to reckon he had "more pressing matters" for focus on than New Foundations for Mathematical Physics [15]. Jheald (talk) 20:03, 8 February 2012 (UTC)
In 1991 Hestenes is writing [16] (p. 15):

Representations of conformal groups in Clifford algebras have been discussed by many authors, notably Angles [11], Lounesto and Latvamaa [12], Ahlfors [13], Maks [14], and Crumeyrolle [15]. The present study aims to show that the conformal split idea simplifies algebraic manipulations, clarifies geometric meanings, and reveals connections among alternative approaches. A complete, efficient and systematic treatment of the conformal group should be helpful in applications to physics. The most thorough discussion of the physical significance of the conformal group for spacetime has been given by Kastrup [16].

(I assume that's the hat-doffing you were referring to?)
But I can't remember, does Hestenes give an approach in that 1991 paper that's equivalent to that in Hestenes (2000), or equivalently Hest/Sob (1984) ? Jheald (talk) 20:17, 8 February 2012 (UTC)
The cap doffing on page 20 " It has been studied by Angles [11], but without the conformal split to simplify the analysis." is also in relation to the equation on page 19. In fact those and subsequent pages seem to be a reprise and further development of the work in Hest/Sob. (Of course it all looks so easy with hindsight...)Selfstudier (talk) 23:14, 8 February 2012 (UTC)
We can add this into the mix as well; in Udo Hertrich-Jeromin, Introduction to Möbius differential geometry (2003) sub section "I.1.4 The Clifford Algebra approach" (p. 5) the refs 154 and 155 are to the above mentioned Hestenes Design paper and to his paper with Ziegler on projective GA.
I will quote the whole bit:

"I.1.4 The Clifford algebra approach. After describing Möbius geometry as a subgeometry of projective geometry it is rather natural to use the Clifford algebra of the (n+2)-dimensional space of homogeneous coordinates of the host projective space, equipped with a Minkowski scalar product, in order to describe geometric objects (cf., [155] and [154]) — just notice that the development of this algebra, initiated by H. Grassmann in [130] and [133] and by W. K. Clifford in [77], was originally motivated by geometry, as its original name, “geometric algebra,” suggests. For example, the description of spheres of any codimension becomes extremely simple using this approach."

— Preceding unsigned comment added by Selfstudier (talkcontribs) 00:28, 9 February 2012 (UTC)
A useful book, for putting the various isomorphic models that can represent conformal geometry into context and into relation with one another. Jheald (talk) 13:10, 10 February 2012 (UTC)
That 1999 Li Paper effectively gives credit to Wachter for the plane model (seems reasonable in view of the history we have found so far) and then goes on to describe "The Wachter model of the Space" which I think maybe is not so reasonable, I would call that the classical or projective model of Mobius geometry which somehow got lost in the wash in favour of more convoluted versions involving Lie Geometry/6D/Vahlen matrices etc.(Actually, the Hestenes tweaked version appears still to be unknown or unrecognized among much of the mathematical community or perhaps it is just not "interesting" enough for them)Selfstudier (talk) 12:52, 9 February 2012 (UTC)
It's probably worth our having a small section towards the end of the article intorducing the Vahlen matrices as the most direct generalisation of the original Möbius mapping in C, per Lounesto (1997, 2001), Hestenes (1991) and the Hertrich-Jeromin book chapters 6 & 7 (2003). They follow on naturally, having introduced CGA. Jheald (talk) 13:23, 10 February 2012 (UTC)
Being a real GAer, I won't have anything to do with C or matrices (other than in implementations perhaps) and will leave that to others:-)Selfstudier (talk) 13:55, 10 February 2012 (UTC)

## Pin Spin Spinor Groups Stuff

Our article Möbius transformation certainly discusses the identification of

"the group of transformations SO+(1,3) ... with the group PSL(2,C) of Möbius transformations of the sphere, by exhibiting the action of the spin group on spinors (Penrose & Rindler 1986)"

and, later,

"In summary, the action of the restricted Lorentz group SO+(1,3) agrees with that of the Möbius group PSL(2,C). This motivates the following definition. In dimension n ≥ 2, the Möbius group Möb(n) is the group of all orientation-preserving conformal isometries of the round sphere Sn to itself. By realizing the conformal sphere as the space of future-pointing rays of the null cone in the Minkowski space R1,n+1, there is an isomorphism of Möb(n) with the restricted Lorentz group SO+(1,n+1) of Lorentz transformations with positive determinant, preserving the direction of time."

The contribution of GA would seem to be to make all of this much much more concrete. Jheald (talk) 18:42, 8 February 2012 (UTC)

This is a bit complicated, I agree that these "groupy" abstractions are useful on occasion but just at the moment it is going to take us far from where we are, we can and should return to it once we have the main skeleton of the article in placeSelfstudier (talk) 15:08, 9 February 2012 (UTC)

I will also mention here that the Hest/Sob 1984 treatment/demonstration is a much ~cleaner explanation (of CGA) than the above generalisation; the spinor aspect is also explicit.Selfstudier (talk) 14:45, 10 February 2012 (UTC)

I was just meaning this as a further data point for the discussion above of how much the isomorphism
C(r,s) ≅ O(r+1,s+1) [Hestenes 2009 [17], eqn. 16]
-- i.e. between the conformal group in Rr,s and the orthogonal group in Rr+1,s+1 -- was already well known and well appreciated.
so eg also the statement in chapter 1 of Hertrich-Jeromin's book (2003), introducing the 'classical' theory of Möbius transformations, at sec 1.3.14 (p. 46--47), that:

Theorem: Any Möbius transformation μ ∈ Möb(n) comes from a Lorentz transformation μ ∈ O1(n+2). This Lorentz transformation is unique up to a sign.

How long has this been absolutely commonplace in the subject? That would seem to be something that is useful to know. Our other article gives a cite from Penrose & Rindler going back to 1986. But was what they were writing commonplace even then?
It also was meant to suggest that, once this article is done, it might be appropriate to go to the article Möbius transformation and edit in cross-references to here in appropriate places in that article, and this would seem to be one of them. Jheald (talk) 15:37, 10 February 2012 (UTC)
OK, I see what you are driving at now, good idea.Selfstudier (talk) 15:46, 10 February 2012 (UTC)
I have an idea this is going to hard to pin down, Penrose was into this stuff back in the 60's in a twistor context. You could in principle get the result from Liouville's 1850 theorem(s) for n>2, I suspect people have just simply piled successive results on top of one another and used what they need to support whatever context they happened to be working in. I will keep looking, however, although it seems to me that historically, "groupy" stuff came after not before. (FWIW, just my opinion, it was probably known a long time ago and has probably been rediscovered in different ways for different contexts ever since)Selfstudier (talk) 19:22, 10 February 2012 (UTC)
This sort of thing.... Conformal Transformations and Clifford Algebras Pertti Lounesto and Esko Latvamaa 1980 (also at JSTOR) Selfstudier (talk) 19:33, 10 February 2012 (UTC)
Li also makes this "well known" claim and cites 4 sources, the earliest being 1987 (Berger); anyway, I have these sources but since Li unhelpfully did not specify any page numbers, I will have to hunt them down.Selfstudier (talk) 20:59, 10 February 2012 (UTC)
Does Vahlen get close to it, back in 1902? Or is it at least later than that? Jheald (talk) 21:15, 10 February 2012 (UTC)
This paper, by Doran and the Lasenbys (2002), seems close to suggesting it goes all the way back to Möbius:

The solution is provided by the conformal model of Euclidean geometry, originally introduced by Möbius in his study of the geometry of spheres. The essential new feature of this space is that it has mixed signature, so the inner product is not positive definite. In the nineteenth century, when these developments were initiated, mixed signature spaces were a highly original and somewhat abstract concept. Today, however, physicists and mathematicians routinely study such spaces in the guise of special relativity, and there are no formal difficulties when computing with vectors in these spaces.

Jheald (talk) 22:06, 10 February 2012 (UTC)
Vahlen IS the same thing but all in disguise as matrices (which are just representations). As for Doran and the Lasenbys, that's what I keep saying, it's Mobius geometry (flat space conformal) but that doesn't necessarily mean the isomorphism (although it wouldn't surprise me). If I had it and I could read German, it wouldn't surprise me if it was in Blaschke (1929) (Mobius bible).
What I don't get is why or how attention suddenly got shifted from 4,1 to Vahlen/6D; the only thing that I can see is that Lie Sphere geometry was/is considered to be a generalization (or includes) both Mobius and Laguerre geometries and so subsequent to that, everyone just concluded that all you had to do was use Lie and you would get Mobius by default; while that's true in principle (you can just knock off the n+3 term in the mapping) it actually doesn't completely work in the detail because a transform in Lie isn't always classical.Selfstudier (talk) 23:23, 10 February 2012 (UTC)
Maybe Klein conformal group of nD Euc iso isometry group of hyperbolic(n+1)1872. He was mostly pally with Lie. Selfstudier (talk) 00:16, 11 February 2012 (UTC)
One other less obvious possibility is Killing who worked in the same area as Lie and introduced the hyperboloid model although I think this might have been later on, I'm thinking the origin was probably with Klein, possibly as a result of collaborating with Lie.Selfstudier (talk) 11:59, 11 February 2012 (UTC)
OK, finally got round to this and tracked it down via one of Li's references, Ratcliffes 2006 Foundations of Hyperbolic Manifolds p 143 which says that the isomorphism "follows immediately from observations of Klein" in his 1872 Ueber Liniengeometrie und metrische Geometrie and 1873 paper Ueber die sogenannte Nicht-Euklidische Geometrie.Selfstudier (talk) 20:39, 1 March 2012 (UTC)

## The basis trick

On page 15 of the Colapinto/VERSOR paper above there is a decent outline explanation of this and some tables which might be handy to include in the construction section, anyone know offhand if I can use the usual LaTeX for that? — Preceding unsigned comment added by Selfstudier (talkcontribs) 20:20, 9 February 2012 (UTC)

You can certainly use WP tables, eg:
 ${\displaystyle n_{\infty }=e_{-}-e_{+}}$ ${\displaystyle n_{o}={\tfrac {1}{2}}(e_{-}+e_{+})}$ ${\displaystyle e_{+}=n_{o}-{\tfrac {1}{2}}n_{\infty }}$ ${\displaystyle e_{-}=n_{o}+{\tfrac {1}{2}}n_{\infty }}$
or
 ${\displaystyle n_{\infty }=e_{-}-e_{+}}$ ${\displaystyle n_{o}={\tfrac {1}{2}}(e_{-}+e_{+})}$ ${\displaystyle e_{+}=n_{o}-{\tfrac {1}{2}}n_{\infty }}$ ${\displaystyle e_{-}=n_{o}+{\tfrac {1}{2}}n_{\infty }}$
Alternatively, you can do the whole thing in LaTeX,
{\displaystyle {\begin{aligned}n_{\infty }&=e_{-}-e_{+}\qquad \qquad &n_{o}&={\tfrac {1}{2}}(e_{-}+e_{+})\\e_{+}&=n_{o}-{\tfrac {1}{2}}n_{\infty }\qquad \qquad &e_{-}&=n_{o}+{\tfrac {1}{2}}n_{\infty }\end{aligned}}}
Jheald (talk) 19:40, 10 February 2012 (UTC)
Great, I will try my hand at those shortly....Selfstudier (talk) 23:26, 10 February 2012 (UTC)
Ok here we go, looks a bit odd but...
 ${\displaystyle e_{+}^{2}=1}$ ${\displaystyle e_{-}^{2}=-1}$ ${\displaystyle n_{\infty }^{2}=0}$ ${\displaystyle n_{o}^{2}=0}$ ${\displaystyle n_{\infty }\wedge n_{o}=E}$ ${\displaystyle n_{o}\wedge n_{\infty }=-E}$ ${\displaystyle e_{+}\wedge e_{-}=E}$ ${\displaystyle n_{\infty }\wedge e_{-}=E}$ ${\displaystyle e_{+}e_{-}=E\,}$ ${\displaystyle e_{-}e_{+}=-E\,}$ ${\displaystyle n_{\infty }n_{o}=-1+E}$ ${\displaystyle n_{o}n_{\infty }=-1-E}$ ${\displaystyle n_{0}\cdot n_{\infty }=-1}$ ${\displaystyle n_{\infty }\cdot n_{o}=-1}$ ${\displaystyle n_{\infty }\cdot E=e}$ ${\displaystyle E\cdot n_{\infty }=-n_{\infty }}$ ${\displaystyle E\cdot n_{o}=n_{o}}$ ${\displaystyle n_{o}\cdot E=-n_{o}}$ ${\displaystyle E^{2}=1\;}$ ${\displaystyle e_{+}\cdot e_{-}=0}$ ${\displaystyle En_{\infty }=-n_{\infty }}$ ${\displaystyle n_{\infty }E=n_{\infty }}$ ${\displaystyle En_{o}=n_{o}\;}$ ${\displaystyle n_{o}E=-n_{o}\;}$
${\displaystyle E=e_{+}\wedge e_{-}}$ is the Minkowski plane.
Hope you don't mind, I've tweaked the above table to centre the equations in each box. Still, as you say, looks a bit odd though. You could line up all the equals-signs if you used Latex; but then you wouldn't get the HTML table grid. I'm not sure we need this in the article -- if anything, we should perhaps aim to reduce the equations that are already in the relevant section (or make them less prominent). Jheald (talk) 15:20, 13 February 2012 (UTC)
I think one would not want to include this entire table in the article. Also any dot product between anything but vectors must also be defined. You should in general be using a left or right contraction, whatever notation you choose to use (and I would suggest the symbols ⌋ and ⌊). Hestenes's "inner product" is often eshewed by other authors. — Quondum 15:46, 13 February 2012 (UTC)
A further thought: if you want to present all this information, it would probably look neater in a set of multiplication tables, one for each operator (geometric product, wedge and left contraction). — Quondum 15:53, 13 February 2012 (UTC)

I wasn't suggesting it be included, just available somewhere and here will do, these identities are very convenient when you get to the stage of actually doing some stuff. The left contraction is already used by everybody as it is the most appropriate for implementations (eliminates cases). Selfstudier (talk) 15:53, 13 February 2012 (UTC)

## "n_o part of r"

Is there better language available than "no part of r" ?

I'm thinking of "no component of r", but it's not really a component, because no and n aren't properly orthogonal. So how does one best replace the word "component" when dealing with a non-orthogonal basis ?

I've also edited expressions like r . no into various parts of the text and the captions, to mean something like the "no part of r". But (I now realise) warning these expressions are quite wrong, because here r . no does not give the "no part of r". So (second question), is there a convenient (and intuitive) mathematical shorthand to indicate the "no part of r" when the no part of r is not r . no ? Jheald (talk) 12:52, 11 February 2012 (UTC)

Update equations now fixed. Jheald (talk) 14:57, 11 February 2012 (UTC)
I guess the word I'm looking for may be "co-ordinate", as distinct from "component". Now, is there a good notation to indicate in formulas the "no coordinate of r" ? Jheald (talk) 13:07, 11 February 2012 (UTC)
Maybe a point has a representation like
a = 1.00*e1 + 0.50*e2 + 0.00*e3 + 1.00*no + 0.625*ni

" you see that in general there are 5 coefficients to the speciﬁcation of a point. The coefficient of no is an overall weight, the coefficients of e1,e2, e3 are its (weighted) position relative to no, and we will explain in section 2.1 that the coefficient of ni is proportional to half the modulus squared of the displacement vector. The{ e1, e2, e3 } part is therefore how you would represent a point by its location vector in the regular representation of Euclidean space, the ‘no’ part extends that to the representation of a point in the homogeneous model, and the ‘ni’ part makes this into the new ‘conformal’ representation."

That is from The GA Viewer tutorial for CGA, does it help? Selfstudier (talk) 13:47, 11 February 2012 (UTC)
-- So "coefficient" could be the missing word I was looking for ?
Anyhow, I've re-worked the text a little, so even if there isn't a good mathematical symbology to indicate the no part or coefficient or co-ordinate, we should now get along well enough without it. Jheald (talk) 14:57, 11 February 2012 (UTC)
Yes, coefficient seems OK although when you are "doing the math" so to speak it is not usually necessary to descend from the algebra as such....Selfstudier (talk) 15:00, 11 February 2012 (UTC)

Going back to a question earlier in this thread about naming of "parts" of a vector – in particular, terminology appropriate to non-orthonormal bases. My impression (from the appropriate Wikipedia articles) is that

• "coefficient" means the scalar multiplier of the basis element in the expression of the vector
• "component" means the vector parallel to the basis element in the expression for the vector
• "coordinate" is a poor term to use: it seems to apply to orthonomal Cartesian coordinates.

Thus, if v = viei,

vj = ejv are the coefficients in the ei basis, and
viei are the components in that basis,

this being general for any basis, including the null basis of CGA. It follows that −n is the covector associated with the no basis vector, and −no is the covector associated with the n basis vector. I suggest we use the terms coefficient and component in this way. We could devote a section to the basis and this explanation. — Quondum 15:29, 12 February 2012 (UTC)

Think he maybe wants to distinguish between Cartesian components and whatever these components are....Selfstudier (talk) 16:33, 12 February 2012 (UTC)

## Pics

I might upload a somewhat different picture that will clarify things a bit more, perhaps.....Selfstudier (talk) 15:40, 11 February 2012 (UTC)

Good, but too busy, I think. Too much going on for one diagram.
I think what might maybe be useful would be to have two pix.
I think keep something very similar to what we have at the moment as a second pic. (But a new re-drawing would be good, perhaps based very closely on what you've just uploaded, so that (i) it's an SVG and doesn't have the horrible JPG noise, and (ii) it doesn't look as if we've just cut-and-pasted it from fig 2.3 in the Hestenes/Li/Rockwood paper, which the present pic rather unfortunately does).
But in addition to that, I think add a first picture, showing the x axis stereographically projected to a circle in the ${\displaystyle \mathbf {e} _{x}\wedge \mathbf {e} _{+}}$ plane, where that plane has the same orientation as the plane in the second picture (and being careful to show/imply infinity being mapped to the correct place). Jheald (talk) 17:10, 11 February 2012 (UTC)
Incidentally, we might for the time being leave out the word "horosphere" -- I'm not sure it adds very much to understanding; its proper place is perhaps at most a footnote. Jheald (talk) 17:54, 11 February 2012 (UTC)
Heh, yes, you can find that pic (or variants) in quite a few places (It "belongs" to Li originally).Selfstudier (talk) 17:55, 11 February 2012 (UTC)
Wachter will be very upset if the horosphere isn't in there and I don't agree that it doesn't add to understanding, on the contrary understanding that properly (remember the useful identities that were deleted in the other article?) simply explains the whole thing without having to go through all the palaver of a non-linear embed (which is anyway pretty obvious from the pic I uploaded, at least to a reader that has made it that far and is unlikely to be casual).88.82.206.110 (talk) 18:07, 11 February 2012 (UTC)Selfstudier (talk) 18:07, 11 February 2012 (UTC)
Here is another one, but I would have to check whether we can use this without permission (the other one is OK) YACGAPICSelfstudier (talk) 19:10, 11 February 2012 (UTC)
That's an impressively crazy diagram. Lines arrows circles and letters whirling about in all directions! Where on earth did you find it? (Incidentally, I think their R1 is pointing in the wrong direction). Also, I'm not quite sure, is P meant to be a curved surface, that the horosphere parabola is being etched on, or is it the usual no=1 ?
But honestly, I do think that simpler is better. That's why I think something very similar to what we have at the moment is probably best (though maybe angled so that the n axis points a little further round towards us, and the x axis is angled round a little further away from us, so more square on.
And then above it, with all the axes in the same orientation, I'd put a second diagram, showing the stereographic projection -- perhaps with the x axis in green mapping to a circle in blue; and then the same blue circle (translated) appearing in the diagram below, being mapped to a blue parabola.
Would breaking it up in two (related) steps like that not be a better way to avoid cognitive overload? Jheald (talk) 20:07, 11 February 2012 (UTC)
I think it should be remembered that the pictures (and any other pictures) are representing a thing which is non-Euclidean and that we already know that CGA is projective, to me the important thing is the final outcome not the intermediate steps (which seems something more like some sort of mathematical verification but short of a proof) For myself, I won't object if you would like to do pictures in the way you are suggesting because you think that they will aid in comprehension and I will put a different kind of picture in the GA article, that way we can cover both sides of the discussion.Selfstudier (talk) 20:45, 11 February 2012 (UTC)

I had forgotten what an effort I always find it to try to make good graphics. I've now got a reasonable skeleton drawing in R for the geometry of the mapping F; only to find that vector output from R's 3D plotting routines is horribly broken.

I may try translating what I've done into Gnuplot, which apparently does support SVG output, which can then be modified / decorated / adapted in Inkscape. Jheald (talk) 16:16, 12 February 2012 (UTC)

An alternative to the picture route which explains the stereographic projection somewhat more mathematically could be SterProjEquiv which has only a simple diagram associated with it (for a laugh, there is also YA even more fantastic pic for CGA:-)Selfstudier (talk) 23:36, 12 February 2012 (UTC)
Work in progress
I'm starting to get there -- at least for an SVG analogue of the existing JPG.
The picture to the right shows a raw outline I've got out of Gnuplot. There's still a lot to do in Inkscape -- tighten up the bounding box, thicken the lines, shade the cone, add lettering, generally try to make it look simpler and cleaner, etc. But it's a beginning.
It's fairly easy to turn the raw Gnuplot figure round to other angles, but not once its fixed in SVG. So any suggestions for improvements very welcome -- but preferably soon, before I've got too far into polishing it.
The ref above is interesting. Like Hestenes & Sobczak (1984), it seems to suggest that the whole mapping can be achieved by a single appropriate stereographic projection. I need to look more closely at the equations, at the moment I can't yet see it.
It's because the projective part is only really a scaling so it's "projectively equivalent" subject to the -1 constraint (that's what I mean by saying we already know it's projective).Selfstudier (talk) 10:26, 13 February 2012 (UTC)
You can also go back a few pages to p153 for more intuition.....Selfstudier (talk) 10:41, 13 February 2012 (UTC)
The purple thingy is the basis of the hyperboloid model of hyperbolic geometry (the asymptotic surface is a cone)Selfstudier (talk) 10:53, 13 February 2012 (UTC)
I do propose to make a twin of the current diagram though, with just the green x axis and the first circle highlighted, plus some contruction lines to show how that part of the transformation definitely can be achieved by stereographic projection. Jheald (talk) 01:39, 13 February 2012 (UTC)
At the moment I'm not seeing what I expect. In particular, I expect a vector x in the base space (i.e. in the green axis) to be mapped by F onto the red parabola, and thus to the 1D homogeneous space through that point on the parabola (a line). This process could be shown as a line to a horizontal plane (the addition of no) followed by a horizontal line to the parabola; else simply a vertical plane through x intersecting with the parabola (this is a very simple but accurate visualization of the mapping: I can picture a semitransparent plane through x intersecting a semitransparent truncated double cone (of another colour) with the horosphere inscribed, and a line on the cone through (not just to) the origin). And any picture will have to rely on careful shading to use 3D visual cues.
I've not been able to figure out what the circle and red dot represent. I've obviously missed something.
I've also tried to visualize the reflection −(xe+)n(xe+), but without success. — Quondum 09:01, 13 February 2012 (UTC)
I don't know whether you have tried it, but there is a demo you can run in GA Viewer (DEMOc2ga) which is also good for developing intuition (a static view is like here [18]Selfstudier (talk) 11:03, 13 February 2012 (UTC)
I haven't gone so far as to download GA Viewer yet (I may have tried once, if so, I wasn't successful), but I have Perwass's CLUViz. They all probably need time putting together the math to describe the objects to be vizualized. — Quondum 11:21, 13 February 2012 (UTC)
Yes, that's true but all the demos can just be stepped through, so for simply getting a leg up on (C)GA, GA Viewer is much better than CLUViz whereas if you have time to invest, then CLUViz is a better bet...Selfstudier (talk) 11:56, 13 February 2012 (UTC)
Wrt −(xe+)n(xe+), it may be worth noting that (xe+) is not normalised, so this is not just a simple reflection, there is also a scaling factor which depends on x. When Hestenes & Sobczak (1984) produce this formula (p. 303), they talk about how it can be interpreted by comparison with an expression that they had discussed earlier; but I haven't yet apllied myself to have a look at that. Jheald (talk) 11:53, 13 February 2012 (UTC)
Sorry, it is going to be hard to interpret without a caption, letters marked on it, some arrows & some better shading.
What the pic tries to show is the composition of the mapping in three stages as currently offered in the text.
So first the x axis is taken by stereographic projection, projecting from the point e+ (the red dot), to a circle in the xe+ plane (purple). To this is then added e, which translates that circle onto the null cone (the blue circle). This is then re-scaled, mapping the blue circle to the red parabola. Jheald (talk) 09:19, 13 February 2012 (UTC)
Ahhhh - got it. Yes, the line from the red dot to x intersecting the circle etc. is needed, but you were getting there. You may want to consider my construction as a second diagram: rather obvious from the formula, fewer steps if one uses the idea of the intersecting point of two planes and a cone, the position of one plane determined by x along the green axis. Obviously the fixed plane and cone can be replaced by a simple parabola, but the cone is generated again by the projective map. — Quondum 09:41, 13 February 2012 (UTC)

## Horosphere

Although it says horosphere (that being what it is "really" in 5D) the picture itself is only 3D (2,1) so it is horocycle ie a horocycle lives on a horosphere and is the intersection of a horosphere with a plane through the centre of the horosphere (in 3D hyperbolic geometry). It doesn't matter whether the plane is hyperbolic or Euclidean as we are dealing with spheres and intersection of 2 spheres is a plane. Then the horosphere is the set of points and the horocycles are lines with the Euclidean distance encoded between any 2 points.Selfstudier (talk) 12:19, 12 February 2012 (UTC)

You can see more about this in Ch4 of Sommer from the resources section [19]Selfstudier (talk) 12:38, 12 February 2012 (UTC)

In 4,1 the horosphere is a 3D paraboloid...Selfstudier (talk) 15:08, 12 February 2012 (UTC)
I have now looked at the Wiki for horosphere (redirects to horoball which I assume is a topological generalization) and for horocycle; these entries ought to be combined (and improved); how does one go about doing that? The combining, I mean. Selfstudier (talk) 12:51, 12 February 2012 (UTC)
Copy the material you want to move to the right place in the new article, making sure to say where it came from in the edit summary.
Then replace the whole of the old article which is to be suppressed with #REDIRECT [[New article]]
Also, maybe leave a note on the talk page to say what you've done, and mention the old talk page if there's anything worthwhile on it. Jheald (talk) 13:26, 12 February 2012 (UTC)
OK, I did it, hope it's right....Selfstudier (talk) 14:43, 12 February 2012 (UTC)
The merge is fine; though I suspect the article could use a re-write. My guess is that it should probably be titled Horosphere, and start with an intuitive definition of the term, before any formal technical definition in terms of horoballs. The 2D hyperbolic case should perhaps get its own specific subsection. Jheald (talk) 16:21, 12 February 2012 (UTC)
Yes, I put a cleanup tag on it and I need to improve the article, try to do (at least some of)that shortly...Selfstudier (talk) 16:44, 12 February 2012 (UTC)
I've moved the article to Horosphere, and added a paragraph about Wachter and Lobachevsky. But it could still use some heavy attention from someone to really straighten the article out and make it accessible. (For example, better integrate the definition in terms of normals) Jheald (talk) 18:43, 12 February 2012 (UTC)
Hum, someone came along and removed all the horocycle material....I undid it, we might need a discussion...Selfstudier (talk) 13:20, 13 February 2012 (UTC)
Now they are ganging up, I haven't time for a debate on the merits and will leave them to it, I think...Selfstudier (talk) 15:56, 13 February 2012 (UTC)

## (C)GA "Alternatives"/Motor Algebra

I see there is a ref to the "motor" algebra in mainspace; in this context, Gunn's recent thesis/papers and GA in Practice are more up to date analyses of this. — Preceding unsigned comment added by Selfstudier (talkcontribs) 14:31, 11 February 2012 (UTC)

There must be an editor at Springer who's really keen on this stuff :-) Jheald (talk) 15:04, 11 February 2012 (UTC)
Yes, bit of a roll at the moment, quite a few GA books about (i saw a philosophy! paper about mathematical practice which described GA as a "quiet revolution". This might be true but there have been false dawn before and it is still not clear how to really make progress meanwhile they persist in teaching the cross as a 3D mainstaySelfstudier (talk) 15:09, 11 February 2012 (UTC)
Some slides of a course here [20]. Post grad, Brazil and CS but hey, at least they are starting to teach it.....Selfstudier (talk) 12:55, 13 February 2012 (UTC)

## Multiplicative split

In a few sources Hestenes uses the so called multiplicative split:

${\displaystyle {\mathcal {G}}_{n+1,1}={\mathcal {G}}_{n}\otimes {\mathcal {G}}_{1,1}}$

He then explains that in this split, vectors in ${\displaystyle {\mathcal {G}}_{n}}$ are actually represented by trivectors of the form

${\displaystyle \mathbf {x} =x\wedge E}$

but it's not clear to me what ${\displaystyle x}$ is nor how the whole thing works. In any case he gets the following formula for ${\displaystyle F(\mathbf {x} )}$

${\displaystyle F(\mathbf {x} )=(n_{o}+\mathbf {x} +{\frac {1}{2}}\mathbf {x} n_{\infty })E=n_{o}+\mathbf {x} E-{\frac {1}{2}}\mathbf {x} n_{\infty }}$

if I'm not mistaken. Notice that in this split ${\displaystyle n_{o}\cdot n_{\infty }=1}$, not ${\displaystyle n_{o}\cdot n_{\infty }=-1}$. One of the things that bother me is that ${\displaystyle F(\mathbf {x} )}$ is not a vector anymore, is it? Then I'm not sure it can be said to be on the null cone, for instance.

It'd be great if someone could make a section about this with detailed explanations.--Grondilu (talk) 15:28, 13 October 2015 (UTC)

The multiplicative and the additive versions are both referred to as the conformal split; practice uses both (Dorst has used both, inclusive Hestenes favored version quite recently). The versions are isomorphic so the results should be the same, however there may be advantages to using one version or the other depending on the use case. I am going to try and fix up this page a bit and I will try and deal with this issue at some point while doing that. Selfstudier (talk) 14:47, 30 October 2017 (UTC)