Talk:Möbius transformation

message: 1) Since the argument of the sqrt operator is in general a complex number, the operator must be a complex number sqrt not a real sqrt.
2) Therefore, some mention of normalization needs to be made. That is, where in the complex plane is the branch cut being made.  A usual place is along the negative real axis.  In that case, the polar angle of the sqrt is normalized to +-90 degrees (+- pi/2 radians).
3) Once you have normalized the square root, the two values are the +-sqrt values.  These are points that mirror each other across the origin.  The quadratic equation has only two roots.  With a precisely defined sqrt operator, the standard quadratic formula unambiguously specifies both roots.
4) A geometrically clearer, and therefore preferable form of the quadratic formula is formed when, given f(z)=a*z*z+b*z+c, let p=-b/(2*c) and q=c/a.  Then the roots are given by p +- sqrt(p*p-q). The value p is an extremum of the function.  The roots are symmetrical about p.  In the case where p*p==q you have a double root.  The other form is rubbish from the educational system and muddies the water.  — Preceding unsigned comment added by 96.55.28.32 (talk) 23:05, 24 February 2017 (UTC)
replies: text: What's written in the article seems clear enough to me.  Concern over the doublevaluedness of the complex square root seems confused, since it is both roots we are considering simultaneously.  There is no need whatsoever to introduce a branch cut.  Sławomir Biały (talk) 23:32, 24 February 2017 (UTC)
thread_title: Fixed point formulae are ambiguous

message: Say, does that java applet work for everyone?
:It works for me. Evil saltine 09:02, 17 Dec 2003 (UTC)
thread_title: Applet

message: 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)
replies: text:  I am not sure I agree it is a Borel subgroup. You need to allow diagonal entries too (same problem in main article).85.1.13.169 (talk) 20:24, 5 February 2008 (UTC)
thread_title:  Parabolic transformations 

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

message: 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)}{-c z + 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)
replies: text: Here is formula. --Adam majewski (talk) 09:41, 30 October 2021 (UTC)
replies: text:  Silly to try answer a 15 years old post that is blatantly wrong. D.Lazard (talk) 11:12, 30 October 2021 (UTC)
replies: text:  https://www.mathwords.com/i/inverse_of_a_matrix.htm --Adam majewski (talk) 18:16, 4 November 2021 (UTC)
thread_title:  Inverse funciton wrong! 

message: 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)
replies: text: 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)
thread_title:  critical problems in the article 

message: 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)
thread_title:  quaternions? 

message: Who the heck ersaed the formula for computing the inverse? Luckilly, it's on this talk page. And I had to totally ferret around in order to find the formula for deriving the characteristic constant from the matrix terms - the whole "characteristic constant" section has been erased altogether, and all that's left is a mere mention in section 6.1. It's like someone with a different idea of what these transformations are all about has been erasing stuff.
Could we please lever things of interest to people like me, who need the equations so we can write computer programs? I appreciate that some people feel that the focus should be the beauty and inner meaning of the equations, but some of us want to calculate things. Maybe there's a way we can have both? I would, however, like to thank the eraser for not wiping section 10, despite its being of practical use.
Oh - and wikipedia for some reason has decided that some of my images didn't have copyright, even though I recall filling out that "yes, I generated these bits" pages for each of them. In any case, we need a set that also includes the parabolic cases. —The preceding unsigned comment was added by 124.176.37.159 (talk) 09:15, 17 December 2006 (UTC)
thread_title:  Organisation 

message: There are two further problems I've noticed. Firstly, the division by zero/infinite issue isn't properly handled ( c=0 => f(infinity)=infinity, else f(infinity)=a/c, f(-d/c)=infinity). This doesn't contribute to a proper understanding of the nature of a function (some people may derive from this that, for example, 1/0=infinity).
Also, the general decomposition does not work for c=0. c=0 => d=/=0 by def'n. Then let f1=(a/d)z, f2=z+(b/d), and composed f2of1. I'd change it myself but...I suck with TeX, and I might be missing the point Wrayal 19:26, 29 May 2007 (UTC)
thread_title:  Further problems 

message: isn't this a better link for the Douglas N. Arnold and Jonathan Rogness video in external links? The creators page about the video with both low res youtube version and high res download, in addition to some images. Images which by the way are currently CC'd, but perhaps could be dual licensed for the article, if someone wanted to contact them. but that's probably not necessary. ._-zro tc 03:04, 8 August 2007 (UTC)
thread_title:  Arnold, Rogness  video link 

message: Hi,
"may be performed by performing" seems a bit awkward.Randomblue 18:47, 28 September 2007 (UTC)
thread_title:  awkward phrasing 

message: Is it clear to everyone else what is meant by automorphism of the Reimann sphere? Does it mean invertible complex-analytic? Perhaps that should be said, since the Reimann sphere is not a group and there are many real analytic (as well as only smooth, as well as only continuous) bijections on the sphere. MotherFunctor 21:18, 2 December 2007 (UTC)
thread_title:  automorphism of the Riemann Sphere 

message: Does anybody know a reference for the second sentence in the article?
"Equivalently, a Möbius transformation may be performed by performing a stereographic projection from a plane to a sphere, rotating and moving that sphere to a new arbitrary location and orientation, and performing a stereographic projection back to the plane."
I can write down a proof, but it seems to be a folk theorem.  The (American) complex analysts I've asked haven't been able to find a written reference.  Is this characterization better known outside of the United States, perhaps?  —Preceding unsigned comment added by 128.101.152.151 (talk) 17:59, 4 December 2007 (UTC)
replies: text: I removed the sentence on the grounds that it appeared to be false based on my initial reading of it.  However, if one correctly understands "moving that sphere to a new location", then it is true.  This could be restored if it were made more precise.  I'm not too worried about finding a reference for it, though.  As soon as it is properly stated, I think it can be filed under "obvious".  siℓℓy rabbit (talk) 14:49, 13 August 2008 (UTC)

text: Your mistake was to ask complex analysts. You should ask geometers.Billlion (talk)
There is now a reference for this statement.  See the article "Mobius transformations revealed", by Arnold and Rogness, published in the Notices of the AMS, volume 55, number 10. best, Sam nead (talk) 14:28, 2 March 2012 (UTC)
thread_title:  Reference for Opening Statement 

message: Why does it say in "Fixed Points" that "every Moebius transformation" has two fixed points? The transformation z->z+1, the archetypal parabolic transformation surely has only one fixed point. I think this is on the whole a great page, but I don't think it's very helpful to say this. Alunmw (talk) 17:58, 21 March 2008 (UTC)
replies: text: It's explained in the next sentence.  Fixed points are always solutions of quadratic equations, so there are always two of them provided one takes into account the appropriate notion of multiplicity.  siℓℓy rabbit (talk) 14:40, 13 August 2008 (UTC)
thread_title:  Parabolic fixed point 

message: I changed the definition of Loxodromic transformation, according to the definition in 
G.Jones & D. Singerman "Complex Functions". I believe it is preferred to have four distinct and non-intersecting types of maps.
Paxinum (talk) 09:35, 3 October 2008 (UTC)
thread_title:  Loxodromic/Parabolic 

message: There are a lot of images that were deleted because they weren't tagged as being in the public domain.  Oddly, it seems that only half of the images in each group were deleted, instead of all of them.  Is it possible for someone to re-generate the deleted images?  If they can't be re-generated directly, for consistency per group, could complete batches be generated?  --ΨΦorg (talk) 05:45, 1 December 2008 (UTC)
thread_title:  Missing images 

message: Under the header elliptic transforms, it is stated that
"A transform is elliptic if and only if c"
Shouldn't that be 
"A transform is elliptic if and only if |λ| = 1, λ not equal to 1."?
If λ = 1, then the trace would be equal to 4, in contradiction to the classification.
Nielius (talk) 09:25, 4 March 2012 (UTC)
thread_title:  Elliptic if and only if |λ| = 1? (Minor remark) 

message: Recently, a IP user has added a new subsection called "Symmetry Princple" (sic). The subject is relevant to the section in which it has been inserted. However, he use his own terminology, calling "symmetry" an involution that is usually called "conjugation" and is never called "symmetry" (in geometry, the symmetries are usually isometries or, at least linear, which is not the case here, except when the generalized circle is a line). Thus I have edited this section to use the correct terminology and make the link with harmonic division, which is fundamental here.
The IP user has reverted my edit for the reason that he prefer his own version. I have reverted again for the above reasons. D.Lazard (talk) 10:10, 18 February 2013 (UTC)
replies: text: John B. Conway uses the term "Symmetry Principle" in Functions of One Complex Variable I (p. 51, 3.19). He also defines "symmetric" (p. 50, 3.17) --50.53.43.172 (talk) 04:48, 26 September 2014 (UTC)
thread_title:  Symmetry or Conjugation 

message: Isn't it more conventional to write for (x_0,x_1,x_2,x_3), associate the Hermitian matrix 
:X=\begin{bmatrix}
x_0+x_3 & x_1+ix_2\\
x_1-ix_2 & x_0-x_3
\end{bmatrix}
to be in better agreement with \sigma^\mu etc.? Freeboson {talk | contribs} 22:05, 30 July 2013 (UTC)
thread_title:  Comment on [[Möbius transformation#Lorentz transformations]] 

message: There is a separate article called linear fractional transformation. The redirects similar to "linear fractional transformation" that had pointed to this article have all been changed to point to it. Here is a complete list:
*
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--50.53.43.172 (talk) 02:13, 26 September 2014 (UTC)
thread_title:  several redirects now point to a separate article called [[linear fractional transformation]] 

message: I moved a lengthy digression on Möbius transformations in higher dimensions here, because the term Möbius transformation is also very widely used to refer to conformal maps of the n-sphere.  It is not just in the context of Liouville's theorem.  See, for example, the second volume of Marcel Berger's "Geometry", or R.W. Sharpe's "Differential geometry: Klein's generalization of Cartan's Erlangen program".  Papers of Shigeo Sasaki and Kentaro Yano refer to the Möbius group in this higher dimensional setting.  The term is firmly established in geometry, and so should at least be mentioned in this main article.  The case has been made that attributing these to Möbius is not historical, and the article can also mention that if we have reliable sources on the matter.  12:01, 18 February 2015 (UTC)
replies: text: I think you might have forgotten to sign completely and thoroughly here Sławomir . YohanN7 (talk) 19:54, 6 March 2015 (UTC)
thread_title:  Higher dimensions 

message: Since the parabolic transformations fix only one point, is it really necessary to almost completely ignore this (I did say "almost"), and repeatedly claim that every Möbius transformation fixes two points?
Sure, go ahead and mention that if the fixed points are counted with multiplicity, then the sum of the multiplicities is always 2.  But that is not the number of fixed points for a parabolic Möbius transformation. 
So please don't pretend that 1 = 2. Even three-year-old children know this is false.Daqu (talk) 23:03, 15 August 2015 (UTC)
thread_title:  Why miscount the fixed points? 

message: Substituted at 02:22, 5 May 2016 (UTC)
thread_title: Assessment comment

message: :The article says : "Möbius transformations are (...) also variously named homographies, homographic transformations, linear fractional transformations (...)"
:It seems that a Möbius transformation is a particular type of linear fractional transformation. Indeed, there is another article with title Linear fractional transformation. Marvoir (talk) 10:26, 14 December 2016 (UTC)
thread_title:  "also (...) named (...) linear fractional transformations" 

message: In mathematics, the number 2 means two.  It does not mean "two when counted with multiplicity."
If you want to add the multiplicities of the fixed points together and claim that the sum is always equal to two — that would be true. But as stated, this is simply false:
"Topologically, the fact that (non-identity) Möbius transformations fix 2 points corresponds to the Euler characteristic of the sphere being 2:
:\chi(\hat{\mathbf{C}})=2."
And No, the fact that the previous section redefines "fixed point" as "fixed point with multiplicity" does not justify this. It is a very bad idea to redefine words in an encyclopedia. But then to use the redefined word in another section??? That is a very, very bad idea.2600:1700:E1C0:F340:295B:3FA:6619:54BE (talk) 22:28, 9 November 2018 (UTC)
thread_title:  False statement needs to be replaced