Talk:Möbius transformation

Applet

Say, does that java applet work for everyone?

It works for me. Evil saltine 09:02, 17 Dec 2003 (UTC)

This comes in handy when doing computing, as terms with possible zeroes in the denominator can be multiplied out.

I cut that out.

One can get rid of the infinities by multiplying out by $z_{2}-z_{1}$ and $Z_{2}-Z_{1}$ as previously noted.

I left this in - but there's a problem with this and the previous comment, as stated. The implication that one can happily divide by zero is not good. In fact it is OK here, as can be seen by talking more systematically about homogeneous coordinates. This is implicit in the second comment, which is why I left it for the moment.

The article still needs work, to adapt the imported material.

Charles Matthews 10:27, 23 Dec 2003 (UTC)

I have updated the applet to demonstrate that the cross product is invariant under a transformation. Sweet. Should update the page too, to mention it.

Also: I understyand the inverse pole a little better now. Will update the text re the poles.

Higher dimensions

The Möbius transfomation is not just a two-dimesnional thing. In fact in higher dimensional Euclidean space the Möbius transformations, which are defined by stereographic projection rather than using complex numbers, are the only conformal mappings. Is this worth putting in? I have actually used these in an applied context, and have a paper with a man made of spheres and cylinders Möbius transformed to make the point [1].

What do folk think? Worth mentioning in the article or is it two long already? Or should be somewhere else (conformal transformations in differential geometry or something) 14:32, 6 Sep 2004 (UTC)

I'm not convinced that linear fractional transformations in more variables are normally called Möbius transfomations. They are certainly of interest. The point about the conformal group in higher dimensions is already mentioned (at conformal geometry?). It would be well worth expanding on that, somewhere.

Charles Matthews 14:41, 6 Sep 2004 (UTC)

There should be a discussion somewhere of the real Mobius transformation on the n-sphere, or equivalently, on the one-point compactification

$R^{n}\cup {\infty }$. This is indeed called a Mobius transformation, see for instance Beardon, The Geometry of discrete groups. Gene Ward Smith 06:21, 21 April 2006 (UTC)

So, you could try amplifying conformal transformation. I've made a conformal geometry category.

Charles Matthews 14:49, 6 Sep 2004 (UTC)

more

I'm fiddling with this User:Pmurray_bigpond.com/Geometry_of_Complex_Numbers but it's not anywhere near ready yet. If it ever is, I might promote it to a topic "Geometry of Complex Numbers (book)".

Lorentz group

Hi all, very nice article, but you've been forgetting something important! Namely the connection with the Lorentz group To be precise, the proper isochronous Lorentz group SO⁺(1,3) is isomorphic as a Lie group to the Möbius group PSL(2,C). This is terribly important in physics, and adds interest to the mathematics of Möbius transformations, which are admittedly already sufficiently interesting to deserve a long article devoted to them.

I just added a bunch of really good citations and some discussion of the physical interpretation of three of the conjugacy classes. (Wanted: pictures of parabolics.)

Congragulations to whoever added the illustrations of individual Lorentz transformations--- beautiful! But I think animations of continuous flows would be even more vivid. Or at least figures showing the flow lines for typical elliptic, hyperbolic, loxodromic, parabolic transformations.

I like the fact that the article tries to explain the derivation of the classification of conjugacy classes, but right now the organization seems a bit try. I think this can literally come alive if someone takes the trouble to add animations.

Curiously, except in the comments I just added, the one-parameter subgroups are referred to as "continuous iteration". Since the rest of the article is written at a fairly high level of precision, I feel it is probably worth rewriting this language in terms of one-parameter subgroups.

I use the term "continuous iteration" because I am a computer programmer, not a mathematician. My original interest was in animating trees laid out in hyperbolic space in response to movements of the mouse. See the inxight website. 203.10.231.231 05:03, 10 August 2005 (UTC)

I'd like to bring out more clearly the fact that in this article we are classifying elements of the M group up to conjugacy, but this is essentially the same as classifying the one-parameter Lie subalgebras up to conjugacy. More generally, I'd like to add some discussion of Lie subalgebras to this article as well as to the Lorentz group article, including a nifty graph of the lattice of subalgebras. I'd like to say a bit about the interpretation of the coset spaces (homogeneous spaces) in terms of Kleinian geometry:

• the coset space of the four dimensional subgroup of similitudes on the euclidean plane, which is the stabilizer of the origin in the Moebius action, and the stabilizer of a null line in the Lorentz action, corresponds to conformal geometry of the sphere,

• the coset space of the three dimensional subgroup E(2), the isometry group of the euclidean plane, which is the stabilizer of a null vector in the Lorentz action, corresponds to the momentum space of massless particles (see Wigner's second little group), and is none other than the degenerate geometry of the light cone. (The stabilizer of a null vector is a subgroup of the stabilizer of a null line, and conformal geometry on the sphere can be regarded as a simplified model of the geometry of the light cone.)

• the coset space of the three dimensional subgroup SO(3), which is the stabilizer of a timelike vector in the Lorentz action, corresponds to the momentum space of a massive particle, which is none other than hyperbolic space H³ (see Wigner's first little group).

In general, since these two articles (which were unlinked until today!) are discussing the same thing, it seems fair to put the more mathematical discussion in this article, and the physical discussion in the other article. But Penrose's interpretation of Möbius transformations is so vidid that I think it is justifiable to have a bit of overlap. I might add pictures of the flow lines of the one-parameter subgroups to the Lorentz group article, which I would draw using Maple. Hmmm... actually, I know how to produce the kind of animated pictures I was asking for above, but I'm not sure if I can massage them into a form which will run on the Wiki server. Any suggestions?

This article is already getting rather long, so I'll see if I can explain clearly but concisely the connection between the action on Minkowski spacetime and the action on the Riemann sphere in the other article. ---CH (talk) 2 July 2005 03:57 (UTC)

P. S. Why do I want to see pictures of parabolic transformations? These are very important in both math and physics. In physics, they are called "null rotations" and are needed for new articles related to Petrov classification of exact solutions of Einstein's field equations.---CH (talk) 2 July 2005 04:07 (UTC)CH

P. P. S. Another important and interesting theme is the way that the Moebius group arises as the point symmetry group of an ordinary differential equation. This also has close connections with Kleinian geometry. ---CH (talk) 2 July 2005 19:53 (UTC)

Yes well, seeing that the article is getting rather long, it might be more appropriate to think about how a general intro might be written, directing the user to other articles that review assorted Lorentz gp representation bits and pieces. Having been the last to make major edits here, my apologies on the lack of mention of SO(3,1); oddly enough, I have an entire book devoted to SL(2,C) (although not in the title): Moshe Carmeli Group Theory and General Relativity; rephrases most of GR using SL(2,C). Never liked it much, I admit, but maybe its time to re-read/skim it again. For what its worth, I've yet to see a book on modular forms or Riemann surfaces that even hints at "Lorentz" or physics anywhere in the book. There's oodles to be said on this topic; note also that the interplay of representations is the launching point for supersymmetry. What we really need to do is to think about how such an obviously broad topic can be split into bite-sized chunks. -- linas 3 July 2005 04:31 (UTC)

On reviewing your edits, it seems that one possible new article might be physical interpretation of Mobius transforms which reviews the the various physical, intuitive correspondances between the various forms and the physical realizations of corresponding transforms. Maybe the graphics should be moved there as well. I'm already finding the current article to be rather unbearably long. linas 3 July 2005 05:09 (UTC)

Naming Conventions

I've encountered some different terminology. Sadly, mathematical definitions aren't as standard as we'd all like, so I thought I'd put it out there to figure out if this is at all standard or if I'm crazy.

I've heard the term "fractional linear transformation" or "linear fractional transformation" for functions of the form $z\mapsto {\frac {az+b}{cz+d}}$, and the term "Möbius transformation" for those of the form $z\mapsto {\frac {z-a}{1-{\overline {a}}z}}$; that is, those that, combined with rotations, constitute the fractional linear transformations mapping the unit disc to itself. This is the terminology used by Mathworld.

Also, is there another term which specifically refers to those of the form $z\mapsto {\frac {z-a}{1-{\overline {a}}z}}$? If so, an article should be written.

Usually, Möbius transformations means the same thing as a fractional linear transformation. Either refers to any automorphism of the Riemann sphere. I think the Mathworld article is wrong (or at least unconventional). Someone please correct me if this is not the case.

To the best of my knowledge we have no article on the automorphism group of the unit disc, which is the group PSU(1,1) given by transformations of the form

$z\mapsto {\frac {az+b}{{\bar {b}}z+{\bar {a}}}},\qquad |a|^{2}-|b|^{2}=1$

nor on the automorphism group of the upper half plane, which is the group PSL(2,R). Of course, these two groups are isomorphic and we should probably devote an article to them. I'm not sure what the best name would be. -- Fropuff 7 July 2005 05:12 (UTC)

Stellar movement

There is stuff in this article on how transformations with fixed points directly "ahead" and "behind" correspond to the affect of accellerations in space on where the stars appear to be. What about if the fixed points are not ahead and behind? Presumably that is the situation when you rotate about an axis that is not the same as the axis of "boost".

In general, what path will a point in space, with an initial velocity, appear to follow when an observer accelerates and spins around an axis? Stars are a special case, when the "points" are infinitely far away. What shapes will geometrical objects (planes, lines, spheres) be distorted into?

Given that straight lines remain straight lines under lorentz contraction, and that the klein and poincare models do the same thing at the border of the unit circle, I suppose that things would would be distored in a manner similar to the way the klein-model isometries distort things (only in 3d, not 2).

projective transformations

I found the projective transformations section to be a bit odd. Then I worked out what was going on: the original was written in simpler terms, and then someone had repeated the same thing in more formal terms. After puzzling out that π was referring to the mapping introduxed in the first sentence and that much of the formalism was simply saying the same thing again, I have reorganised the paragraph to integrate the two streams of thought a bit more closeley.

It needs reviewing by someone who understands math (g). It looks liek π is being defined twice, because we use "π:". In fact, π is defined by the words "let us call this mapping π", and the two expressions are not defining it but merely saying something about it. I'm not sure how you "say" this in <math> language.

edited class representative for parabolic

I edited the "class reporesentative" for parabolic transforms to be z+a. This is because the only "pure" parabolic transform (ie, k=0 without anything else) is the identity transform, but putting that in would not really help anyone.

Pretty pictures

Whew! I have bean meaning to do this for some time, and now its done. I have added some 3d pictures of transformations being stereographically projected. The "loxodromic/arbitrary" ones are inaccurate .... but I'm leaving it as-is for the time being because the params are consistent with those in the other pictures.

Parabolic transformations

This article is great! It answered some questions I needed answers to. One problem: it refers to "the subgroup of parabolic transformations". Contrary to what this suggests, the set of all parabolic transformations is not a subgroup! Parabolic transformations of the form

${\begin{pmatrix}1&a\\0&1\end{pmatrix}}$

form a subgroup, which is indeed a Borel subgroup. Heck, I guess I'll go and fix this. John Baez 17:19, 7 March 2006 (UTC)

Fraktur

Is the use of fraktur for matrices standard in the literature? Since it is not elsewhere... Dysprosia 08:11, 15 May 2006 (UTC)

: it's used in The Geometry of Complex Numbers

TeX is a Pain in the Ass

how to format this in TeX? (is wikipedia using TeX or LaTeX?

f1[Z]:= Z+D/C                  (translation)
f2[Z]:= 1/Z                    (inversion and reflection)
f3[Z]:= - (A D-B C)/C^2 * Z    (dilation and rotation)
f4[Z]:= Z+A/C                  (translation)

in parlticular, what i want is a block with 2 vertical alignments, and, the second column should not be in math format.

thx.

Xah Lee 22:44, 25 July 2006 (UTC)

I'd probably use wiki markup mixed with TeX. Something like this:

$f_{1}(z):=z+{\frac {d}{c}}$	(translation)
$f_{2}(z):={\frac {1}{z}}$	(inversion and reflection)
$f_{3}(z):=-{\frac {ad-bc}{c^{2}}}z$	(dilation and rotation)
$f_{4}(z):=z+{\frac {a}{c}}$	(translation)

Actually, MediaWiki software uses neither TeX nor LaTeX, but a strange approximation called texvc. There is an effort underway to switch to new software called blahtex, which supports more of LaTeX and can output MathML. For better or worse, TeX markup is the familiar standard within the mathematics and science community. Try to appreciate it; it could be worse! Consider the MathML markup for a quadratic formula:

   <math xmlns="http://www.w3.org/1999/xhtml">
     <mfrac>
       <mrow>
         <mrow> <mo>-</mo><mi>b</mi> </mrow>
         <mo>±</mo>
         <sqrt>
           <msup><mi>b</mi><mn>2</mn></msup>
           <mo>-</mo>
           <mrow> <mn>4</mn><mo>&it;</mo><mi>a</mi><mo>&it;</mo><mi>c</mi> </mrow>
         </sqrt>
       </mrow>
       <mrow> <mn>2</mn><mo>&it;</mo><mi>a</mi> </mrow>
     </mfrac>
   </math>

All this just to produce

${\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.$

The TeX version is orders of magnitude more user-friendly (and brief!). --KSmrq^T 23:12, 31 July 2006 (UTC)

hi KSmrg, actually i have some personal beef with the entirety of TeX. I think, it has done huge damage to the math community. I did some rant that touched the gist of my views http://xahlee.org/Periodic_dosage_dir/t2/TeX_pestilence.html , which i hope to clean up and add proper format and references in the near future... In fact, despite the verbosity of MathML, i think it is far, far better alternative than TeX simply because that it started with the right basis. Xah Lee 02:29, 1 August 2006 (UTC)

Inverse funciton wrong!

the inversion function on the page is wrong!

the inverse of mobius transformation is actually: (d z - b)/(-c z + a).

The expression for the inverse function as given on the page is actually correct only if a d - b c = {1,0}

i've moved the wrong inverse expr from main page to below:

« with the following two special cases:

• the point $z=-d/c$ is mapped to $\infty$
• the point $z=\infty$ is mapped to $a/c$

We can have Möbius transformations over the real numbers, as well as for the complex numbers. In both cases, we need to augment the domain with a point at infinity.

The condition ad − bc ≠ 0 ensures that the transformation is invertible. The inverse transformation is given by

$z\mapsto {\frac {\left({\frac {d}{ad-bc}}\right)z-\left({\frac {b}{ad-bc}}\right)}{-cz+a}}$

with the usual special cases understood. »

also, the statement about where the points -d/c is mapped to should be addressed somewhere else, as a detail about the mobius transformation. It is not “with these special cases”. Rather, if we want to state that there, it should be something like “MF has 2 points that are notworthy”. Also, we don't “need” to “augment the domain with a point at infinity”... any, my point is that this block of phrasing are terrible.

Xah Lee 08:35, 27 July 2006 (UTC)

critical problems in the article

i like to point out a few things in this article that i think needs fix.

• the angle-preservation property, which is a fundamental property of the MT, is never discussed in the article (other than saying it is a conformal map). It needs at least some discussion on how or why. (the proof of it can be deferred to the circle inversion page)

• the decomposition of MT into simpler affine transformations + circle inversion is not mentioned in the article. This is critically important, as it gives insight of what MT really is from geometric transformation point of view, and the circle inversion is most fundamental key to the whole MT business. This also needs to be mentioned.

• that the inverse function of MT given in the page is incorrect, as i've talked about before. I have given the correct formula in my last edit.

These issues, i tried to correct in my last sequence of edits (see here: http://en.wikipedia.org/w/index.php?title=M%C3%B6bius_transformation&diff=66701420&oldid=66351090 ) I'm not sure a full revert is necessarily a proper course of action, even if my formating and presentation style can be improved.

Some of the style and presentation as they currently are, in my opinion, suffers from jargonization symptom as in most texts, as well as from wikip's collective editing nature. But regardless, it is important to get the above critical math contents present and correct, however or whoever does it.

I hope people will correct these problems. Thanks.

Xah Lee 02:15, 1 August 2006 (UTC)

Unlike some people I don't think reverting major edits is a good way of accomplishing anything positive. As such I've reverted Oleg's revert. --MarSch 10:22, 19 September 2006 (UTC)

quaternions?

Just as the (proper) conformal transformations of S^2 are (P)SL(2, C) also known as Moebius transformations, the (proper) conformal transformations of S^4 are (P)SL(2, H), where H stands for the quaternions and not for the complex upper half-plane. The name "Moebius transformations" probably applies, but fractional linear transformations certainly also applies to the quaternionic variant. The article is very biased towards the complex case. --MarSch 10:46, 19 September 2006 (UTC)