Talk:Möbius transformation

4138:

should adopt for WP, and examples of individual scholars' use is of no value for this. I understand the risks of use of other articles to draw conclusions; however, if we were to make a change in one article without updating all the other articles accordingly, such a change would produce a confusing mess of broken cross-referencing in WP. It is far less problematic to consistently settle on a particular meaning of a term in WP than to use it differently in different articles (unless there is detailed disambiguation in each article, which is itself not worth the price). Accordingly, we need to be very sure about the dominance of a term before changing it everywhere or anywhere. In this article's case, the change would not be to change the article content, but to change the title to "Möbius transformation of the complex numbers", "Complex Möbius transformation", or similar. An example: the term "ring" originated to mean what is

4418:, which is simpler that a pure geometric approach in many ways, so I see its attraction. One can do as you say, in the the sense of defining it geometrically, but then switching to the complex projective line for the bulk of the article. This is less of a rewrite than the pure geometric approach, but does suggest keeping the mindset that this represents a real geometry, and relating back to the geometric interpretation everywhere. Editing the article to match this mindset would take some work, and I'm not signing up for this. — 84: 74: 53: 2725:), and let's say for some reason farmers needed to do this calculation all the time, and that the farmers have a special kind of table that they used exclusively for doing this calculation, and it was so important that every tractor has this table inscribed on the dashboard. If that were the case, I think it would be entirely appropriate for the quadratic residue article to say that it's used in farming and to show the table that the farmers use for doing the calculation. -- 2569: 22: 489:). I think Charles' point is that "Mobius transformation" will in many cases refer to linear fractional transformations of the Riemann sphere. I agree with that although I also think that a good many mathematicians think of a Mobius transformation as something more general. I was certainly surprised to see the general case is not even discussed here. Seems odd. -- 3310: 2093:• 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. 636:

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?

4262:

I think we should add another sentence or two at the end of the lead about how the same kinds of transformations of the Euclidean plane (not complex numbers per se) are also called Möbius transformations – I believe these are indeed what Möbius originally studied, though I am not an expert – and that

4227:

We don't have to "force" consistent terminology, but we should strive to make links coherent with concepts, not blind to differences of the intended meaning. Anything that links to this article should mean to reach this content. Anything that means the "higher dimensional Möbius transformations" of

3994:

You can't treat Knowledge text (of any of these articles) as reliable here. It was written by some random person who never bothered to do any kind of survey, and was most likely also not an authority on these subjects. It's also tricky to take any other individual sources as authorities, as they very

2510:

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

2347:

2 Möbius transformations are defined just as it says. Of course to include dilations we must allow a choice of stereographic projection. For a reference see Kulkarni R S and Pinkall U 1988 Conformal Geometry (Braunschweig: Vieweg). p 95. I used these maps in my paper, jou may like to look at picture

2175:

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

1124:

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

1088:

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

1080:

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

173:

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

3949:

The extension to a higher number of dimensions (in the sense of a projective transformation) seems somewhat dubious, and I'm not sure Ahlfors even meant it as transformations of a higher-dimensional space: from the introduction (which is all that I can see), his reference to 2×2 matrices of Clifford

3135:

I think that because the sentence you quote as the title of this section is expressed poorly, you may have mistaken what that sentence is attempting to say. My guess is that they were only trying to say that each element of the Möbius group (which of course, as a linear fractional transformation, is

2232:

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,

2016:

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

684:

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

510:

in the introduction. The introduction is now in contradiction with most of the article, which uses "Möbius transformation" to refer only to what the introduction now calls "Möbius transformation of the plane". I think this needs to be more systematically integrated into the article, but I don't know

4514:

My understanding is that de Sitter spacetime is a Lorentzian manifold of uniform sectional curvature with the same metric signature (1, 3) or (3, 1) as the flat Minkowski spacetime, and likewise for anti de Sitter spacetime, but either can be embedded as a hypersurface in some flat pseudo-Euclidean

4400:

That is, I think it's better to initially think of "Möbius transformation" as meaning a transformation of the plane rather than thinking of it as a complex function. In just the same way we'd define "translation" as or "reflection" as a geometrical concept rather than starting out with how they can

4204:

I disagree: often it is problematic to attempt to force consistent use of terminology across Knowledge articles. This is especially true in mathematics, where often the terminologies used by mathematicians, physicists and engineers all disagree, yet we need to write articles that work for everyone,

4144:

in WP, despite extensive historical and even modern literature that calls the latter "ring". However, WP has settled on "ring" having an identity element. Since this was settled, working on articles using these terms has become far less frustrating and time-consuming, and reading them is far less

697:

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

635:

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

579:

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

3298:

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

2711:

There is a certain Mobius transformation with an important practical application (electrical engineering); and people in this application area have a graphical method that they use all the time for calculating this Mobius transformation. That seems quite on-topic to me. (I was half-hearted earlier

2324:

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

1097:

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

3119:

I just changed the phrasing of this claim. It's not true that any map is homotopic to the identity on a connected space; see the antipodal map on S^(2). Someone more knowledgeable than me should explain *why* the connectedness of this group shows that any map is homotopic to the identity (I don't

2179:

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

2106:

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.

4518:

I don't know anything about AdS/CFT correspondence, but it is true that the conformal transformations of hyperbolic 3-space can be related to conformal transformations of the 2-sphere, which can be thought of as the boundary of 'ideal points'. I can't tell you whether AdS/CFT correspondence is a

4492:

I could find nothing. I note that de sitter and AdS are usually associated with a metric like (2,3) (see Dirac) or (2,4) (see Lasenby), whereas 3D hyperbolic space has the metric (1,3). Plausibly there is an interesting connection, but obviously this would need to be demonstrated explicitly in a

4174:. Ahlfors is merely one (possibly dated) data point of the statistics that are to be gauged. For example, if the predominant term for a broader class is "linear fractional transformation" or "projective transformation", then a simple comment in the related articles to the effect that "the term 2280:

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

352:

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

4413:

While I have no in-principle objection to the article with this title describing the said transformations of the one-point compactification of the Euclidean plane, and agree that this is likely more correct. Yet, changing to this geometric perspective would seem to imply rewriting most of this

4208:

In the case of the present article, I think we should have an indication of the existence of the "Higher dimensions" article in the lead section. It is reasonably common to hear people say "the only conformal transformations in dimension 3 or higher are Möbius transformations" and these are not

4137:

gets used by some with various (not necessarily compatible) broader meanings. This is likely true for most mathematical terms, but this is not a reason to include every meaning that has ever been used for each. The point is to determine the dominant modern interpretation of the term as what we

3862:

I tend to agree with you. My change was made because the emphatic definition followed by the indication that the terms were equivalent implied a restriction of all the terms to the narrow definition was clearly nonsense. I expect that terminology is not consistent in the literature, i.e. that

4105:(who calls it a "Vector-Transformation", in German which I do not read). But we can also extend the idea to transformations of some other kinds of multivectors if you like, and this has been done in other people's papers, e.g. you can find papers about "Möbius transformations" of Quaternions. – 3844:

Emphatically stating that "Möbius transformations" only apply to the complex plane seems at least somewhat misleading to readers. It's fine to have an article focused on that narrower topic, but someone should try to do a bit of literature survey and make sure that the text accurately reflects

2460:

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

546:

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

169:

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.

1084:

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?

4338:

I think I would actually change the basic definition to say that Möbius transformations are the such-and-such transformations of the Euclidean plane , and that if we identify points in the plane with complex numbers, these can be represented as fractional linear transformations of the form

3976:

No it is not (that I have seen) ever the same as a general "conformal transformation" if you include the 1–2 dimensional cases, where conformal transformations have significantly more flexibility. Möbius transformations in this extended use would always refer only to conformal projections

553:

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.

561:

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.

2781:

Wouldn't it be better for Smith charts to have its own article? Then it would be entirely appropriate to add something about who Smith was, when they invented the charts, what problem this was devised to solve, what kinds of situations they are used in today (in the age of software),

1133:

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

2212:

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 =:

713:

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.

4263:

the same idea generalizes naturally to higher dimensions. We already have a section "Higher dimensions" discussing this, though it could probably be expanded. It also may be worth discussing the orientation-reversing transformations at greater length in the article. –

165:

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).

2090:• 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) 4427:

I think a more significant rework of this page is worth doing for the benefit of (especially non-expert) readers, but I'm not going to tackle that right away either. I'd like to get to it eventually after a big pile of other related projects on other mathematical

2298:"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." 474:

I think this point is worth considering again. The Beardon text describes Mobius transformations on R^n and goes on to show that many of the results in C generalize. To start off the article saying Mobius transformations are transformations of C is misleading.

3919:

suggests that the primary use of "Möbius transformation" applies only to the complex case, that any use of the term as generalized to any other ring may be sporadic, and that "linear fractional transformation" is the preferred term in the more general case.

3913:, where the variables are all elements of any chosen ring. That some authors have extended chosen to extend the term "Möbius transformation" to the rings other than the complex numbers is understandable, but examples are not evidence of a norm. The text at 626:

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 H (see Wigner's first little

2183:

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.

3966:

In all, I am now of the view that WP should present a Möbius transformation strictly as a linear fractional transformation of the complex numbers. That is, we should delete the qualification "of the complex plane" in the first sentence of the lead.

3062: 2008: 1580:, 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: 196:

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.

4223:

The picture I'm getting is that there are two uses that need to be highlighted, and this article should point out that it is about the transformations of the complex numbers; a hatnote might also point out where other uses of the term might be

557:

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.

4187:

I don't think the title needs to change. I just think the text should make sure to be a bit less emphatic that this is "the" meaning of the term, since readers commonly take such claims in Knowledge as gospel. To take your ring example, see

3958:

is an isomorphism rather than an automorphism of projective spaces, which is strictly more general (thus, no-one would claim that "Möbius transformation" is synonymous with "homography" unless they meant it in the very general sense of a

1101:

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

1115:

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.

1057: 1725:, 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. 567:

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.

2301:

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?

3708: 511:

enough about how the term is usually used. If the comment above by C S is correct, the article should perhaps also mention that the term is sometimes used to refer only to the two-dimensional form and sometimes more generally.

324:

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

4485:

The bottom of this article has a mention of the connection between MTs and hyperbolic space. This is agreed-upon mathematics that you can find with some googling. A citation should probably be added but I don't have time.

2440:

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.

3977:(automorphisms) of the full space which map all circles to circles (where "circle" includes straight lines) and preserve cross ratios, of the type that can be obtained by sphere inversions (including reflections). Cf. 3528:, 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 4096:

if you take the domain to be vector-valued and the coefficients to be multivector-valued ("Clifford numbers"). Ahlfors himself wrote a few papers about this topic, one or more of which you might be able to look at:

2674:

But the subject is not really Möbius transformations — rather it is electricity. I think this topic belongs in some closely related article that is mainly about electricity, or possibly it deserves its own article

2646:

is a famous depiction of a Möbius transformation -- it's a complex unit circle for a parameter called Γ, but the circle is filled with markings and labels so that for any Γ you can immediately read off the value of

672:

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.

639:

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.

4451:

That makes sense. As I understand it, the Riemann sphere is a graphical representation of the complex projective line, but this does not imply that they are synonymous, whereas the article sort of mixes them.

2505:

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

1188: 729:

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.

650:

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

3767: 3845:

prevailing usage. (The text might say e.g. "the term Möbius transformation is also regularly applied to more general settings, but this article will focus on transformations of the complex plane.") Ping @

911: 845: 4547:. There has been enough written about these that we could definitely put together an article about it, and I think it would be quite helpful to readers trying to understand the 4-dimensional examples. – 2414:

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.

1700: 2651:=(1+Γ)/(1-Γ). It's used ubiquitously in high-frequency electrical engineering. Maybe this image has a place in this article somewhere? (I'm not sure where it would go, or whether it's too off-topic.) 3769:. There's some literature taking this view, especially in the context of the (somewhat tricky) topic of defining notions of convergence for infinite continued fractions. Among articles on Knowledge, 3867:). I suggest that we "sidestep" the issue here until someone adequately surveys the literature for prevailing usage by introducing the article by saying that only the Möbius transformations over 3533: 3180: 1496: 4489:

It also had an enigmatic sentence after this description, which was "This is the first observation leading to the AdS/CFT correspondence in physics". It had no citation, so I started googling.

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

786: 140: 4395: 4094: 1557: 1359: 4275:

I've added a mention of generalization in the lead; if you want to make it clearer that the transformations of any Möbius geometry can also be called "Möbius transformations", feel free.

2911: 3863:

various authors use the term differently. Such terminology variation is pervasive in mathematics. It makes sense to settle on a primary terminology in WP (which should be researched;

2379:

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,

1414: 584:

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:

4493:

peer-reviewed paper before being cited on wikipedia. And for sure if studies hyperbolic space "leads to" AdS/CFT in any sense, there should be a citation of an introductory textbook.

2747:

Fair enough. My only reason for mentioning farming is that the application of arithmetic to another thing is something that is principally about the other thing, not about arithmetic.

2541: 3282:

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

433: 925:. 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. 4025: 1721:

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

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That just goes to show what I (don't) know ... it is ages since I looked at this stuff, and then even then it was fleeting. Please tweak the article to your satisfaction. —

359:

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)

318: 278: 2399:

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.

1843: 4027:"? I can tell you: he means conformal circle-preserving transformations in n-dimensional Euclidean space. All of which can be represented by transformations of the form 3988:

is to refer to what we might call "circular transformations" of the Euclidean plane, without necessarily considering the transformed points to be complex numbers per se.

1815: 3981:. In 3+ dimensions this is the only kind of conformal transformation you can have, but in the plane there are infinitely many other kinds of conformal transformations. 1795: 550:

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.)

2812:. "Mentioning" it here is exactly what I proposed and then did. You can check the article - it's one figure plus a two-sentence caption with a link. Nothing else. -- 1098:

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.

1871: 847:; that is, those that, combined with rotations, constitute the fractional linear transformations mapping the unit disc to itself. This is the terminology used by 656: 4515:

space of higher dimension in the same way a curved 2-sphere with intrinsic signature (2, 0) can be embedded in 3-dimensional Euclidean space of signature (3, 0).

4294:

seems complex. The 'higher dimension' section assumes a positive-definite metric, which probably does not seem to do justice to the topic, even as a mention. —

689:; 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 3209:. 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: 3421:

The article says : "Möbius transformations are (...) also variously named homographies, homographic transformations, linear fractional transformations (...)"

3338:

point, is it really necessary to almost completely ignore this (I did say "almost"), and repeatedly claim that every Möbius transformation fixes two points?

2684:

I say this for the same reason that if some arithmetical calculation shows something about, say, farming, that would not belong in the article on arithmetic.

935: 2266:

when decomposing the Möbius transformation in four simple funtions, wouldn't it be nice to specify, for funtion 3, the scaling ratio and the rotation angle?

930:

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

4574: 130: 2764:

Möbius transformations; it is about electricity. The article is for information about Möbius transformations. In this article, it would be appropriate to

2176:

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.

3954:, not of the vector space over which the Clifford algebra is defined. Note also that a Möbius transformation is a map onto the input space, whereas a 3578: 341:

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.

2721:

What would be the farming analogy? I think the right analogy would be: Suppose there is a slightly-obscure arithmetical calculation (say, calculating

2187: 2214:

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

2096:• 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. 4163:. I suggest that at this point, we should solicit the opinion of a fairly broad group of editors who might have familiarity with the field (at 2122:

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. --

485:

I think there needs to be a disambiguation notice at the top, explaining where to find the material on higher dimensions (there's some stuff at

106: 3395:

Could do with some orginisation of the page, currently seems to be rather bitty with no overal page low. As ever more history of the subject.

605:

in the Lorentz action, corresponds to the momentum space of massless particles (see Wigner's second little group), and is none other than the

4569: 3864: 2309: 1067:). 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. -- 3424:

It seems that a Möbius transformation is a particular type of linear fractional transformation. Indeed, there is another article with title

398:

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

2191: 162:

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.

3537: 3264: 3184: 2545: 1229: 685:

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

601:

the coset space of the three dimensional subgroup E(2), the isometry group of the euclidean plane, which is the stabilizer of a null

4235:"), aside from perhaps adding another hatnote. I don't have objections to making it seem less emphatic; do you have suggestions? — 4231:

jacobolus, I'm not sure what you would like to change to make it less emphatic (it already qualifies it as "a Möbius transformation

2103:) I'm not sure a full revert is necessarily a proper course of action, even if my formating and presentation style can be improved. 1206: 181: 97: 58: 1140: 2768:

that Möbius transformations are used in Smith charts, but with a link to the article that contains the details about Smith charts.

710: 3995:

often are focused on one particular thing and will give it a name, even while other authors use the same name for something else.

3922: 3717: 3455:

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

3425: 3206: 3199: 368: 4439:(with the former focused on interpretations that are directly spherical, and the latter also including other interpretations). – 3389:, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section. 3104: 857: 791: 4511:

Thanks. A claim like this should definitely have a source, especially since it's not precisely clear what the claim is saying.

2157:

stands for the quaternions and not for the complex upper half-plane. The name "Moebius transformations" probably applies, but

3849:, who just made a change here about the relationship between Möbius transformations and linear fractional transformations. – 3770: 2518:

The IP user has reverted my edit for the reason that he prefer his own version. I have reverted again for the above reasons.

2211:

There are two further problems I've noticed. Firstly, the division by zero/infinite issue isn't properly handled ( c=0 =: -->

1636: 353:

these in an applied context, and have a paper with a man made of spheres and cylinders Möbius transformed to make the point

204: 3813:

The current state of this Knowledge page does not (in my non-expert experience) accurately reflect the usage of the term

3290:'s "Geometry", or R.W. Sharpe's "Differential geometry: Klein's generalization of Cartan's Erlangen program". Papers of 534:

but it's not anywhere near ready yet. If it ever is, I might promote it to a topic "Geometry of Complex Numbers (book)".

3552: 1281:

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

33: 4244:

Möbius transformations in 3+ dimensions aren't different enough from those in 2D to need their own article, unlike the

3555:; how are Möbius transformations related to continued fractions? I couldn't find any mention in the article about it. — 1425: 733:

I've heard the term "fractional linear transformation" or "linear fractional transformation" for functions of the form

2398:

Why does it say in "Fixed Points" that "every Moebius transformation" has two fixed points? The transformation z-: -->

2017:“need” to “augment the domain with a point at infinity”... any, my point is that this block of phrasing are terrible. 3475: 1205:

I am not sure I agree it is a Borel subgroup. You need to allow diagonal entries too (same problem in main article).

4170:

To repeat my point: primarily we need to determine a definition for use in WP, and secondarily this should be the

736: 592:

on the euclidean plane, which is the stabilizer of the origin in the Moebius action, and the stabilizer of a null

2240: 232:

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

4539:

could also be better. I'd also like to see articles about the 2-dimensional analogs, which could e.g. be titled

4342: 3304: 2180:

both? I would, however, like to thank the eraser for not wiping section 10, despite its being of practical use.

198: 4030: 3136:

a holomorphic map S → S) is in the same path-component as the identity, since there is only one path-component.

3125: 2581: 2507: 2313: 1507: 1309: 457: 2844: 2835: 2195: 1880:, as well as for the complex numbers. In both cases, we need to augment the domain with a point at infinity. 1370: 4436: 4415: 3833:

or conformal transformations of 3+ dimensional Euclidean or pseudo-Euclidean spaces (e.g. by Ahlfors (1986)

3268: 2549: 2068: 2037: 1233: 326: 2702:

I actually added it to the article a while ago (after no one had objected for a long time). It's there now.

2161:

certainly also applies to the quaternionic variant. The article is very biased towards the complex case. --

1750:

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

1210: 542:

Hi all, very nice article, but you've been forgetting something important! Namely the connection with the

185: 4544: 3778: 3057:{\displaystyle X={\begin{bmatrix}x_{0}+x_{3}&x_{1}+ix_{2}\\x_{1}-ix_{2}&x_{0}-x_{3}\end{bmatrix}}} 2421: 2332: 2003:{\displaystyle z\mapsto {\frac {\left({\frac {d}{ad-bc}}\right)z-\left({\frac {b}{ad-bc}}\right)}{-cz+a}}} 372: 3991:

Many sources also include transformations which are orientation reversing among "Möbius transformations".

3121: 569: 4501: 4117:

Looking at the papers citing Ahlfors might also give you somewhere to start. But here's another example

3286:-sphere. It is not just in the context of Liouville's theorem. See, for example, the second volume of 652: 476: 463: 403: 388: 335: 39: 3817:

in academic literature. I have seen this term regularly used for linear fractional transformation over

2491: 174:

you have a double root. The other form is rubbish from the educational system and muddies the water.

83: 4098: 2404: 4540: 4536: 4001: 3465: 3400: 3176: 2817: 2730: 2659: 2629:(bottom left), and the corresponding value of Γ (bottom right), related by the Möbius transformation 2591: 2305: 2101:

http://en.wikipedia.org/search/?title=M%C3%B6bius_transformation&diff=66701420&oldid=66351090

1225: 440: 177: 4145:

confusing. Similar approaches have been taken with respect to adoption of a convention for defining

21: 4551: 4523: 4443: 4405: 4315: 4267: 4245: 4196: 4164: 4125: 4109: 3853: 3464:

Topologically, the fact that (non-identity) Möbius transformations fix 2 points corresponds to the

3100: 3070: 2480:

Under the header elliptic transforms, it is stated that "A transform is elliptic if and only if c"

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on Knowledge. If you would like to participate, please visit the project page, where you can join

4308: 4189: 3978: 2523: 2400: 2384: 2372: 2357: 2282: 2215: 2064: 2051: 2033: 1259: 486: 380: 89: 4291: 4287: 2669:

This looks very interesting, and surely Smith charts should be described somewhere in Knowledge.

283: 243: 73: 52: 3946:

are complex numbers (in which case the transformation is also called a Möbius transformation)".

3890:

Looking at this a bit more, a linear fractional transformation is a transformation of the form

1722: 4453: 4419: 4324: 4295: 4236: 4179: 3968: 3878: 3818: 3774: 3433: 3319: 2722: 2622: 2577: 2512: 2446: 2415: 2326: 1822: 677: 663: 644: 224: 2325:

for it, though. As soon as it is properly stated, I think it can be filed under "obvious".

344:

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

4497: 3963:", and the restriction of "Möbius transformation" to complex numbers is affirmed there too). 3830: 2914: 2625:

with coefficient Γ. Each point on the Smith chart simultaneously represents both a value of

1800: 1572:

Actually, MediaWiki software uses neither TeX nor LaTeX, but a strange approximation called

1248:

Is the use of fraktur for matrices standard in the literature? Since it is not elsewhere...

694: 516: 2029: 1763: 4532: 4282:; I have no idea whether he extended it. As a geometry it has naturally been extended to 4253: 4214: 3396: 2813: 2726: 2655: 2352:. It is a long time since I wrote that paper though so I am not about to fix the articel. 2267: 2258: 1891:≠ 0 ensures that the transformation is invertible. The inverse transformation is given by 848: 376: 4304: 4283: 4279: 2760:

To elaborate on my main point: The use of Möbius transformations in a Smith chart is not

1222:

I agree that this isn't a Borel ... I changed this in the text ... hope you don't mind.

1193:

form a subgroup, which is indeed a Borel subgroup. Heck, I guess I'll go and fix this.

2568: 2483:

Shouldn't that be "A transform is elliptic if and only if |λ| = 1, λ not equal to 1."?

2346:

So the article as it stands only defines Möbius transformations in the plane. For n: -->

1848: 1052:{\displaystyle z\mapsto {\frac {az+b}{{\bar {b}}z+{\bar {a}}}},\qquad |a|^{2}-|b|^{2}=1} 4554: 4548: 4526: 4520: 4505: 4496:

All this points to it appears to be someone's original research, so I have removed it.

4456: 4446: 4440: 4432: 4422: 4408: 4402: 4327: 4318: 4312: 4298: 4270: 4264: 4257: 4239: 4218: 4199: 4193: 4182: 4128: 4122: 4112: 4106: 3971: 3881: 3856: 3850: 3800: 3782: 3564: 3541: 3437: 3404: 3369: 3365: 3323: 3272: 3188: 3145: 3141: 3129: 3109: 3096: 2821: 2791: 2787: 2734: 2693: 2689: 2663: 2553: 2533: 2527: 2495: 2470: 2462: 2450: 2429: 2408: 2388: 2361: 2340: 2317: 2285: 2270: 2261: 2244: 2218: 2199: 2165: 2126: 2116: 2072: 2055: 2041: 2023: 1729: 1710: 1290: 1252: 1249: 1237: 1214: 1197: 1194: 572: 520: 498: 479: 443: 208: 189: 2486:

If λ = 1, then the trace would be equal to 4, in contradiction to the classification.

2060: 4563: 4140: 3960: 3796: 3560: 3386: 3291: 3287: 2519: 2380: 2368: 2353: 2349: 2113: 2047: 2020: 1726: 1287: 581: 543: 494: 360: 3360:

So please don't pretend that 1 = 2. Even three-year-old children know this is false.

4153: 3846: 3429: 3315: 3295: 2442: 2145:) also known as Moebius transformations, the (proper) conformal transformations of 1068: 690: 674: 667: 660: 641: 3877:

are considered here, but that various generalizations exit with names such as . —

3703:{\displaystyle a_{0}+{\cfrac {b_{0}}{a_{1}+{\cfrac {b_{1}}{a_{2}+{_{\ddots }}}}}}} 3412:

Last edited at 17:59, 14 April 2007 (UTC). Substituted at 02:22, 5 May 2016 (UTC)

3791:

That is true; I hadn't seen that connection before. Thanks for the explanation. —

329:. This is implicit in the second comment, which is why I left it for the moment. 3822: 3570:

I have no particular opinion on the categorization, but, to answer the question:

2809: 2643: 2573: 2561: 2162: 2123: 1877: 1063:

nor on the automorphism group of the upper half plane, which is the group PSL(2,

512: 354: 102: 4209:

sufficiently distinct objects to merit being discussed in a separate article. —

3834: 3115:"Next, the Möbius group is connected, so any map is homotopic to the identity." 4249: 4210: 3955: 3915: 1707: 715: 699: 79: 3950:

numbers makes me assume he is referring to fractional linear transformations

2461:

directly, for consistency per group, could complete batches be generated? --

4159: 3361: 3137: 2783: 2685: 1263: 504:

In the meantime, two additions concerning higher dimensions have been made:

854:

Also, is there another term which specifically refers to those of the form

435:. This is indeed called a Mobius transformation, see for instance Beardon, 2229: 1706:

The TeX version is orders of magnitude more user-friendly (and brief!). --

4414:

article, which is written from the perspective of transformations of the

3792: 3556: 2234: 2233:

if someone wanted to contact them. but that's probably not necessary. ._-

2099:

These issues, i tried to correct in my last sequence of edits (see here:

2087:

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

709:

On reviewing your edits, it seems that one possible new article might be

490: 4205:

or sometimes we need to write separate articles for different audiences.

3303:

I think you might have forgotten to sign completely and thoroughly here

1747:

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

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in the Lorentz action, corresponds to conformal geometry of the sphere,

2295:

Does anybody know a reference for the second sentence in the article?

3998:

What do you think Ahlfors might mean by a "Möbius transformation of

4099:

https://scholar.google.com/scholar?q=ahlfors+Möbius+Transformations

3357:

the number of fixed points for a parabolic Möbius transformation.

2367:

Your mistake was to ask complex analysts. You should ask geometers.

2567: 1573: 2046:

Silly to try answer a 15 years old post that is blatantly wrong.

1296:

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

4119:

https://akjournals.com/view/journals/10998/14/1/article-p93.xml

4102: 1576:. There is an effort underway to switch to new software called 1183:{\displaystyle {\begin{pmatrix}1&a\\0&1\end{pmatrix}}} 332:

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

15: 1275:

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

788:, and the term "Möbius transformation" for those of the form 4228:

a space should link to a different but suitable destination.

3762:{\displaystyle \textstyle z\mapsto a_{k}+{\frac {b_{k}}{z}}} 1723:

http://xahlee.org/Periodic_dosage_dir/t2/TeX_pestilence.html

617:, and conformal geometry on the sphere can be regarded as a 3809:

Clarity about the scope of the name "Möbius transformation"

3169:

imply that all Möbius transformations are homotopic to the

3158:

is connected (and says nothing about the space it acts on).

1753:

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

3416:"also (...) named (...) linear fractional transformations" 906:{\displaystyle z\mapsto {\frac {z-a}{1-{\overline {a}}z}}} 840:{\displaystyle z\mapsto {\frac {z-a}{1-{\overline {a}}z}}} 3714:

can be viewed as a composition of Möbius transformations

3198:

several redirects now point to a separate article called

1695:{\displaystyle {\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.} 383:?). It would be well worth expanding on that, somewhere. 240:

One can get rid of the infinities by multiplying out by

3660: 3640: 3618: 3598: 3381: 3256: 3249: 3242: 3235: 3228: 3221: 3214: 2100: 609:

geometry of the light cone. (The stabilizer of a null

508: 505: 4345: 3721: 3664: 3644: 3622: 3602: 3299:

sources on the matter. 12:01, 18 February 2015 (UTC)

2937: 1262:. He uses them for modius transformations and for 2*2 1149: 375:. They are certainly of interest. The point about the 4178:

is used by some authors to refer to this concept". —

4033: 4004: 3720: 3581: 3478: 3073: 2925: 2847: 2257:"may be performed by performing" seems a bit awkward. 1900: 1851: 1825: 1803: 1766: 1639: 1510: 1428: 1373: 1312: 1143: 938: 860: 794: 739: 406: 286: 246: 2712:

but I'm feeling more confident now that it belongs.)

588:

the coset space of the four dimensional subgroup of

532:

User:Pmurray_bigpond.com/Geometry_of_Complex_Numbers

101:, a collaborative effort to improve the coverage of 2110:I hope people will correct these problems. Thanks. 2061:

https://www.mathwords.com/i/inverse_of_a_matrix.htm

4389: 4088: 4019: 3761: 3702: 3510: 3154:The quote and title of this section says that the 3086: 3056: 2905: 2679:it is adequately explained, not just illustrated). 2350:http://www.iop.org/EJ/abstract/0266-5611/14/1/012/ 2137:Just as the (proper) conformal transformations of 2002: 1865: 1837: 1809: 1789: 1694: 1584:<math xmlns="http://www.w3.org/1999/xhtml": --> 1551: 1491:{\displaystyle f_{3}(z):=-{\frac {ad-bc}{c^{2}}}z} 1490: 1408: 1353: 1182: 1106:. I'm not sure how you "say" this in <math: --> 1081:axis that is not the same as the axis of "boost". 1051: 905: 839: 780: 427: 312: 272: 3448:In mathematics, the number 2 means two. It does 631:In general, since these two articles (which were 3835:"Clifford Numbers and Möbius Transformations in 3345:the fixed points are counted with multiplicity, 1258:it's used in "The Geometry of Complex Numbers", 3511:{\displaystyle \chi ({\hat {\mathbf {C} }})=2.} 3385:, and are posted here for posterity. Following 2544:). He also defines "symmetric" (p. 50, 3.17) -- 2476:Elliptic if and only if |λ| = 1? (Minor remark) 3916:Homography § Homographies of a projective line 507:in the section on Lorentz transformations and 379:in higher dimensions is already mentioned (at 219:Say, does that java applet work for everyone? 3379:The comment(s) below were originally left at 3334:Since the parabolic transformations fix only 2836:Möbius transformation#Lorentz transformations 1744:the inversion function on the page is wrong! 781:{\displaystyle z\mapsto {\frac {az+b}{cz+d}}} 657:exact solutions of Einstein's field equations 8: 4190:Ring (mathematics) § Notes on the definition 4133:You don't need to convince me that the term 3773:is probably the one that comes the closest. 2348:of a man who has been Möbius transformed! 1876:We can have Möbius transformations over the 711:physical interpretation of Mobius transforms 4519:direct higher-dimensional analog of this. – 4390:{\textstyle z\mapsto {\frac {az+b}{cz+d}},} 4311:rather than anything Möbius did directly. – 3926:confirms this: "In the most basic setting, 3452:mean "two when counted with multiplicity." 2013:with the usual special cases understood. » 1266:matrices that he uses to represent circles. 4101:Ahlfors credits the idea to Vahlen (1902) 4089:{\displaystyle x\mapsto (ax+b)(cx+d)^{-1}} 3984:Another (perhaps even more common) use of 3174: 1552:{\displaystyle f_{4}(z):=z+{\frac {a}{c}}} 1354:{\displaystyle f_{1}(z):=z+{\frac {d}{c}}} 613:is a subgroup of the stabilizer of a null 460:. I've made a conformal geometry category. 175: 47: 4352: 4344: 4077: 4032: 4011: 4007: 4006: 4003: 3747: 3741: 3732: 3719: 3683: 3679: 3670: 3665: 3650: 3645: 3637: 3628: 3623: 3608: 3603: 3595: 3586: 3580: 3488: 3486: 3485: 3477: 3078: 3072: 3040: 3027: 3015: 2999: 2985: 2969: 2957: 2944: 2932: 2924: 2906:{\displaystyle (x_{0},x_{1},x_{2},x_{3})} 2894: 2881: 2868: 2855: 2846: 1952: 1914: 1907: 1899: 1855: 1850: 1824: 1802: 1779: 1765: 1658: 1652: 1640: 1638: 1539: 1515: 1509: 1477: 1454: 1433: 1427: 1396: 1378: 1372: 1341: 1317: 1311: 1144: 1142: 1111:edited class representative for parabolic 1037: 1032: 1023: 1014: 1009: 1000: 981: 980: 963: 962: 945: 937: 887: 867: 859: 821: 801: 793: 746: 738: 420: 411: 405: 304: 291: 285: 264: 251: 245: 2841:Isn't it more conventional to write for 1756:« with the following two special cases: 1409:{\displaystyle f_{2}(z):={\frac {1}{z}}} 913:? If so, an article should be written. 3551:This article is listed in the category 3459:. But as stated, this is simply false: 49: 19: 4401:be represented as complex functions. – 3534:2600:1700:E1C0:F340:295B:3FA:6619:54BE 3181:2601:200:C082:2EA0:189E:F594:5A2E:F75E 3120:think it follows from connectedness). 2536:uses the term "Symmetry Principle" in 371:in more variables are normally called 2639:I made this image a while ago --: --> 7: 3444:False statement needs to be replaced 3307: 95:This article is within the scope of 3382:Talk:Möbius transformation/Comments 3205:There is a separate article called 2538:Functions of One Complex Variable I 687:Group Theory and General Relativity 621:of the geometry of the light cone.) 428:{\displaystyle R^{n}\cup {\infty }} 38:It is of interest to the following 4575:High-priority mathematics articles 3827:Geometry of Möbius Transformations 3532:??? That is a very, very bad idea. 2808:The article about Smith charts is 2276:automorphism of the Riemann Sphere 1832: 1804: 421: 158:Fixed point formulae are ambiguous 14: 3387:several discussions in past years 3257:Linear fractional transformations 3229:Fractional linear transformations 2159:fractional linear transformations 369:linear fractional transformations 115:Knowledge:WikiProject Mathematics 4020:{\displaystyle \mathbb {R} ^{n}} 3923:Linear fractional transformation 3489: 3426:Linear fractional transformation 3341:Sure, go ahead and mention that 3308: 3243:Fractional-linear transformation 3222:Fractional linear transformation 3207:linear fractional transformation 3200:linear fractional transformation 2083:critical problems in the article 923:fractional linear transformation 118:Template:WikiProject Mathematics 82: 72: 51: 20: 3067:to be in better agreement with 2291:Reference for Opening Statement 437:The Geometry of discrete groups 135:This article has been rated as 4349: 4278:Presumably Möbius studied the 4172:predominant prevailing meaning 4074: 4058: 4055: 4040: 4037: 3771:Generalized continued fraction 3725: 3499: 3493: 3482: 3330:Why miscount the fixed points? 3273:02:13, 26 September 2014 (UTC) 3189:16:09, 20 September 2023 (UTC) 3171:identity Möbius transformation 3161:Since a conncted Lie group is 3087:{\displaystyle \sigma ^{\mu }} 2900: 2848: 2554:04:48, 26 September 2014 (UTC) 2271:18:51, 28 September 2007 (UTC) 2262:18:47, 28 September 2007 (UTC) 2166:10:46, 19 September 2006 (UTC) 2127:10:22, 19 September 2006 (UTC) 1904: 1527: 1521: 1445: 1439: 1390: 1384: 1329: 1323: 1033: 1024: 1010: 1001: 986: 968: 942: 864: 798: 743: 1: 4431:I also think we should split 3438:10:26, 14 December 2016 (UTC) 3130:22:20, 18 November 2013 (UTC) 2528:10:10, 18 February 2013 (UTC) 2515:, which is fundamental here. 2200:09:15, 17 December 2006 (UTC) 998: 456:So, you could try amplifying 209:23:32, 24 February 2017 (UTC) 190:23:05, 24 February 2017 (UTC) 109:and see a list of open tasks. 4570:B-Class mathematics articles 4103:https://eudml.org/doc/158052 3553:Category:Continued fractions 3542:22:28, 9 November 2018 (UTC) 3405:10:33, 2 November 2006 (UTC) 2590:, terminated at a load with 2471:05:45, 1 December 2008 (UTC) 2318:17:59, 4 December 2007 (UTC) 2286:21:18, 2 December 2007 (UTC) 2073:18:16, 4 November 2021 (UTC) 2056:11:12, 30 October 2021 (UTC) 2042:09:41, 30 October 2021 (UTC) 1215:20:24, 5 February 2008 (UTC) 892: 826: 3801:16:06, 5 October 2021 (UTC) 3370:23:03, 15 August 2015 (UTC) 3215:Fractional linear transform 3146:05:02, 16 August 2015 (UTC) 2822:14:06, 16 August 2015 (UTC) 2792:04:53, 16 August 2015 (UTC) 2735:01:17, 16 August 2015 (UTC) 2694:23:10, 15 August 2015 (UTC) 2451:09:35, 3 October 2008 (UTC) 2430:14:40, 13 August 2008 (UTC) 2341:14:49, 13 August 2008 (UTC) 2224:Arnold, Rogness video link 1418:(inversion and reflection) 573:05:03, 10 August 2005 (UTC) 313:{\displaystyle Z_{2}-Z_{1}} 273:{\displaystyle z_{2}-z_{1}} 4591: 3353:is always 2. But that is 2409:17:58, 21 March 2008 (UTC) 2362:16:21, 19 March 2009 (UTC) 2245:03:04, 8 August 2007 (UTC) 2117:02:15, 1 August 2006 (UTC) 1730:02:29, 1 August 2006 (UTC) 1093:projective transformations 921:means the same thing as a 580:article as well as to the 521:08:55, 10 April 2009 (UTC) 480:20:21, 18 March 2007 (UTC) 444:06:21, 21 April 2006 (UTC) 4555:18:53, 8 March 2024 (UTC) 4527:18:43, 8 March 2024 (UTC) 4506:11:17, 8 March 2024 (UTC) 3865:some WP conventions exist 3783:05:38, 9 March 2021 (UTC) 3565:21:15, 9 March 2021 (UTC) 3394: 3324:19:54, 6 March 2015 (UTC) 3110:22:05, 30 July 2013 (UTC) 2664:14:59, 12 July 2013 (UTC) 2600:and normalised impedance 2496:09:25, 4 March 2012 (UTC) 2389:14:28, 2 March 2012 (UTC) 2024:08:35, 27 July 2006 (UTC) 1838:{\displaystyle z=\infty } 1711:23:12, 31 July 2006 (UTC) 1631:All this just to produce 1291:22:44, 25 July 2006 (UTC) 1198:17:19, 7 March 2006 (UTC) 1129:Parabolic transformations 1104:saying something about it 547:article devoted to them. 338:10:27, 23 Dec 2003 (UTC) 134: 67: 46: 4457:23:49, 23 May 2023 (UTC) 4447:23:10, 23 May 2023 (UTC) 4423:20:28, 23 May 2023 (UTC) 4409:19:36, 23 May 2023 (UTC) 4328:21:35, 22 May 2023 (UTC) 4319:20:39, 22 May 2023 (UTC) 4299:18:15, 22 May 2023 (UTC) 4271:16:34, 22 May 2023 (UTC) 4258:16:18, 22 May 2023 (UTC) 4240:15:15, 22 May 2023 (UTC) 4219:13:13, 22 May 2023 (UTC) 4200:13:10, 22 May 2023 (UTC) 4183:12:44, 22 May 2023 (UTC) 4129:05:28, 22 May 2023 (UTC) 4113:04:59, 22 May 2023 (UTC) 3972:01:23, 22 May 2023 (UTC) 3882:15:24, 21 May 2023 (UTC) 3857:19:48, 20 May 2023 (UTC) 2582:characteristic impedance 2576:. A wave travels down a 2219:19:26, 29 May 2007 (UTC) 1500:(dilation and rotation) 1271:TeX is a Pain in the Ass 1253:08:11, 15 May 2006 (UTC) 1238:15:14, 25 May 2009 (UTC) 680:2 July 2005 19:53 (UTC) 647:2 July 2005 03:57 (UTC) 499:13:30, 1 June 2008 (UTC) 458:conformal transformation 363:14:32, 6 Sep 2004 (UTC) 227:09:02, 17 Dec 2003 (UTC) 141:project's priority scale 4437:complex projective line 4416:complex projective line 4303:Sort of... but article 2501:Symmetry or Conjugation 1810:{\displaystyle \infty } 1740:Inverse funciton wrong! 1071:7 July 2005 05:12 (UTC) 718:3 July 2005 05:09 (UTC) 702:3 July 2005 04:31 (UTC) 666:2 July 2005 04:07 (UTC) 530:I'm fiddling with this 466:14:49, 6 Sep 2004 (UTC) 391:14:41, 6 Sep 2004 (UTC) 367:I'm not convinced that 327:homogeneous coordinates 98:WikiProject Mathematics 4391: 4090: 4021: 3986:Möbius transformations 3825:(e.g. by Kisil (2012) 3763: 3704: 3512: 3468:of the sphere being 2: 3088: 3058: 2907: 2654:Just a thought. :-D -- 2636: 2004: 1867: 1839: 1811: 1791: 1790:{\displaystyle z=-d/c} 1696: 1553: 1492: 1410: 1355: 1184: 1053: 919:Möbius transformations 907: 841: 782: 429: 314: 274: 28:This article is rated 4392: 4176:Möbius transformation 4135:Möbius transformation 4091: 4022: 3829:) or multivectors in 3815:Möbius transformation 3764: 3705: 3573:A continued fraction 3513: 3250:Linear fractional map 3236:Fractional-linear map 3089: 3059: 2908: 2571: 2394:Parabolic fixed point 2190:comment was added by 2005: 1868: 1840: 1812: 1792: 1697: 1554: 1493: 1411: 1356: 1185: 1054: 908: 842: 783: 653:Petrov classification 430: 373:Möbius transfomations 315: 275: 4545:Anti-de Sitter plane 4537:Anti-de Sitter space 4343: 4233:of the complex plane 4031: 4002: 3718: 3579: 3547:Continued fractions? 3476: 3466:Euler characteristic 3071: 2923: 2845: 2436:Loxodromic/Parabolic 1898: 1849: 1823: 1801: 1764: 1637: 1622:⁢</mo: --> 1612:⁢</mo: --> 1608:⁢</mo: --> 1508: 1426: 1371: 1310: 1141: 936: 858: 792: 737: 404: 320:as previously noted. 284: 244: 121:mathematics articles 3952:of Clifford numbers 3663: 3643: 3621: 3601: 1866:{\displaystyle a/c} 4535:is not great, and 4481:Ads/CFT connection 4387: 4309:incidence geometry 4086: 4017: 3979:inversive geometry 3819:hyperbolic numbers 3759: 3758: 3700: 3696: 3691: 3657: 3615: 3508: 3375:Assessment comment 3163:pathwise connected 3084: 3054: 3048: 2903: 2723:quadratic residues 2637: 2000: 1863: 1835: 1807: 1787: 1692: 1549: 1488: 1406: 1351: 1260:Hans Schwerdtfeger 1180: 1174: 1049: 999: 903: 837: 778: 725:Naming Conventions 487:conformal geometry 425: 381:conformal geometry 310: 270: 90:Mathematics portal 34:content assessment 4382: 4288:higher dimensions 3961:conformal mapping 3831:Clifford algebras 3756: 3698: 3693: 3496: 3410: 3409: 3278:Higher dimensions 3191: 3179:comment added by 2623:signal reflection 2578:transmission line 2513:harmonic division 2320: 2308:comment added by 2203: 1998: 1974: 1936: 1687: 1676: 1565: 1564: 1547: 1483: 1404: 1349: 1228:comment added by 993: 989: 971: 901: 895: 835: 829: 776: 348:Higher dimensions 223:It works for me. 192: 180:comment added by 155: 154: 151: 150: 147: 146: 4582: 4396: 4394: 4393: 4388: 4383: 4381: 4367: 4353: 4246:Möbius transform 4095: 4093: 4092: 4087: 4085: 4084: 4026: 4024: 4023: 4018: 4016: 4015: 4010: 3912: 3768: 3766: 3765: 3760: 3757: 3752: 3751: 3742: 3737: 3736: 3709: 3707: 3706: 3701: 3699: 3697: 3695: 3694: 3692: 3690: 3689: 3688: 3687: 3675: 3674: 3662: 3658: 3656: 3655: 3654: 3642: 3638: 3633: 3632: 3620: 3616: 3614: 3613: 3612: 3600: 3596: 3591: 3590: 3517: 3515: 3514: 3509: 3498: 3497: 3492: 3487: 3392: 3391: 3384: 3313: 3312: 3311: 3259: 3252: 3245: 3238: 3231: 3224: 3217: 3108: 3093: 3091: 3090: 3085: 3083: 3082: 3063: 3061: 3060: 3055: 3053: 3052: 3045: 3044: 3032: 3031: 3020: 3019: 3004: 3003: 2990: 2989: 2974: 2973: 2962: 2961: 2949: 2948: 2915:Hermitian matrix 2913:, associate the 2912: 2910: 2909: 2904: 2899: 2898: 2886: 2885: 2873: 2872: 2860: 2859: 2635: 2620: 2426: 2425: 2418: 2337: 2336: 2329: 2303: 2250:awkward phrasing 2237: 2207:Further problems 2185: 2149:^4 are (P)SL(2, 2141:^2 are (P)SL(2, 2009: 2007: 2006: 2001: 1999: 1997: 1980: 1979: 1975: 1973: 1953: 1941: 1937: 1935: 1915: 1908: 1872: 1870: 1869: 1864: 1859: 1844: 1842: 1841: 1836: 1816: 1814: 1813: 1808: 1796: 1794: 1793: 1788: 1783: 1701: 1699: 1698: 1693: 1688: 1686: 1678: 1677: 1663: 1662: 1653: 1641: 1558: 1556: 1555: 1550: 1548: 1540: 1520: 1519: 1497: 1495: 1494: 1489: 1484: 1482: 1481: 1472: 1455: 1438: 1437: 1415: 1413: 1412: 1407: 1405: 1397: 1383: 1382: 1360: 1358: 1357: 1352: 1350: 1342: 1322: 1321: 1304: 1303: 1240: 1189: 1187: 1186: 1181: 1179: 1178: 1125:other pictures. 1076:Stellar movement 1058: 1056: 1055: 1050: 1042: 1041: 1036: 1027: 1019: 1018: 1013: 1004: 994: 992: 991: 990: 982: 973: 972: 964: 960: 946: 912: 910: 909: 904: 902: 900: 896: 888: 879: 868: 846: 844: 843: 838: 836: 834: 830: 822: 813: 802: 787: 785: 784: 779: 777: 775: 761: 747: 695:Riemann surfaces 619:simplified model 464:Charles Matthews 434: 432: 431: 426: 424: 416: 415: 389:Charles Matthews 336:Charles Matthews 319: 317: 316: 311: 309: 308: 296: 295: 279: 277: 276: 271: 269: 268: 256: 255: 236:I cut that out. 201: 123: 122: 119: 116: 113: 92: 87: 86: 76: 69: 68: 63: 55: 48: 31: 25: 24: 16: 4590: 4589: 4585: 4584: 4583: 4581: 4580: 4579: 4560: 4559: 4541:De Sitter plane 4533:De Sitter space 4483: 4368: 4354: 4341: 4340: 4292:Möbius geometry 4073: 4029: 4028: 4005: 4000: 3999: 3891: 3873: 3811: 3743: 3728: 3716: 3715: 3680: 3666: 3661: 3659: 3646: 3641: 3639: 3624: 3619: 3617: 3604: 3599: 3597: 3582: 3577: 3576: 3549: 3530:another section 3474: 3473: 3446: 3418: 3380: 3377: 3349:the sum of the 3332: 3309: 3280: 3255: 3248: 3241: 3234: 3227: 3220: 3213: 3203: 3117: 3095: 3074: 3069: 3068: 3047: 3046: 3036: 3023: 3021: 3011: 2995: 2992: 2991: 2981: 2965: 2963: 2953: 2940: 2933: 2921: 2920: 2890: 2877: 2864: 2851: 2843: 2842: 2839: 2630: 2618: 2611: 2601: 2599: 2589: 2566: 2503: 2478: 2458: 2438: 2423: 2422: 2416: 2396: 2334: 2333: 2327: 2310:128.101.152.151 2293: 2278: 2252: 2243: 2235: 2226: 2209: 2186:—The preceding 2173: 2135: 2085: 2030:Here is formula 1981: 1957: 1948: 1919: 1910: 1909: 1896: 1895: 1847: 1846: 1821: 1820: 1799: 1798: 1762: 1761: 1742: 1679: 1654: 1642: 1635: 1634: 1628: 1626:</mfrac: --> 1511: 1506: 1505: 1473: 1456: 1429: 1424: 1423: 1374: 1369: 1368: 1313: 1308: 1307: 1279: 1273: 1246: 1223: 1173: 1172: 1167: 1161: 1160: 1155: 1145: 1139: 1138: 1131: 1122: 1120:Pretty pictures 1113: 1095: 1078: 1031: 1008: 961: 947: 934: 933: 880: 869: 856: 855: 814: 803: 790: 789: 762: 748: 735: 734: 727: 540: 528: 441:Gene Ward Smith 407: 402: 401: 377:conformal group 350: 300: 287: 282: 281: 260: 247: 242: 241: 217: 199: 160: 120: 117: 114: 111: 110: 88: 81: 61: 32:on Knowledge's 29: 12: 11: 5: 4588: 4586: 4578: 4577: 4572: 4562: 4561: 4558: 4557: 4529: 4516: 4512: 4482: 4479: 4478: 4477: 4476: 4475: 4474: 4473: 4472: 4471: 4470: 4469: 4468: 4467: 4466: 4465: 4464: 4463: 4462: 4461: 4460: 4459: 4433:Riemann sphere 4429: 4398: 4386: 4380: 4377: 4374: 4371: 4366: 4363: 4360: 4357: 4351: 4348: 4336: 4335: 4334: 4333: 4332: 4331: 4330: 4276: 4260: 4229: 4225: 4206: 4202: 4168: 4115: 4083: 4080: 4076: 4072: 4069: 4066: 4063: 4060: 4057: 4054: 4051: 4048: 4045: 4042: 4039: 4036: 4014: 4009: 3996: 3992: 3989: 3982: 3964: 3947: 3885: 3884: 3868: 3841:), and so on. 3810: 3807: 3806: 3805: 3804: 3803: 3786: 3785: 3755: 3750: 3746: 3740: 3735: 3731: 3727: 3724: 3712: 3711: 3710: 3686: 3682: 3678: 3673: 3669: 3653: 3649: 3636: 3631: 3627: 3611: 3607: 3594: 3589: 3585: 3571: 3548: 3545: 3522: 3521: 3507: 3504: 3501: 3495: 3491: 3484: 3481: 3445: 3442: 3441: 3440: 3422: 3417: 3414: 3408: 3407: 3376: 3373: 3351:multiplicities 3331: 3328: 3327: 3326: 3279: 3276: 3261: 3260: 3253: 3246: 3239: 3232: 3225: 3218: 3202: 3196: 3195: 3194: 3193: 3192: 3159: 3149: 3148: 3122:DeathOfBalance 3116: 3113: 3081: 3077: 3065: 3064: 3051: 3043: 3039: 3035: 3030: 3026: 3022: 3018: 3014: 3010: 3007: 3002: 2998: 2994: 2993: 2988: 2984: 2980: 2977: 2972: 2968: 2964: 2960: 2956: 2952: 2947: 2943: 2939: 2938: 2936: 2931: 2928: 2902: 2897: 2893: 2889: 2884: 2880: 2876: 2871: 2867: 2863: 2858: 2854: 2850: 2838: 2832: 2831: 2830: 2829: 2828: 2827: 2826: 2825: 2824: 2799: 2798: 2797: 2796: 2795: 2794: 2774: 2773: 2772: 2771: 2770: 2769: 2753: 2752: 2751: 2750: 2749: 2748: 2740: 2739: 2738: 2737: 2716: 2715: 2714: 2713: 2706: 2705: 2704: 2703: 2697: 2696: 2681: 2680: 2671: 2670: 2616: 2609: 2597: 2587: 2565: 2558: 2557: 2556: 2534:John B. Conway 2502: 2499: 2477: 2474: 2457: 2456:Missing images 2454: 2437: 2434: 2433: 2432: 2395: 2392: 2377: 2376: 2344: 2343: 2292: 2289: 2277: 2274: 2251: 2248: 2239: 2225: 2222: 2208: 2205: 2192:124.176.37.159 2172: 2169: 2134: 2131: 2130: 2129: 2084: 2081: 2080: 2079: 2078: 2077: 2076: 2075: 2011: 2010: 1996: 1993: 1990: 1987: 1984: 1978: 1972: 1969: 1966: 1963: 1960: 1956: 1951: 1947: 1944: 1940: 1934: 1931: 1928: 1925: 1922: 1918: 1913: 1906: 1903: 1883:The condition 1874: 1873: 1862: 1858: 1854: 1834: 1831: 1828: 1817: 1806: 1786: 1782: 1778: 1775: 1772: 1769: 1741: 1738: 1737: 1736: 1735: 1734: 1733: 1732: 1714: 1713: 1704: 1703: 1702: 1691: 1685: 1682: 1675: 1672: 1669: 1666: 1661: 1657: 1651: 1648: 1645: 1627:</math: --> 1625:</mrow: --> 1617:</mrow: --> 1616:</sqrt: --> 1615:</mrow: --> 1601:</msup: --> 1592:</mrow: --> 1585:<mfrac: --> 1583: 1582: 1581: 1569: 1568: 1567: 1566: 1563: 1562: 1561:(translation) 1559: 1546: 1543: 1538: 1535: 1532: 1529: 1526: 1523: 1518: 1514: 1502: 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4156: 4155: 4150: 4149: 4143: 4142: 4136: 4132: 4131: 4130: 4127: 4124: 4120: 4116: 4114: 4111: 4108: 4104: 4100: 4081: 4078: 4070: 4067: 4064: 4061: 4052: 4049: 4046: 4043: 4034: 4012: 3997: 3993: 3990: 3987: 3983: 3980: 3975: 3974: 3973: 3970: 3965: 3962: 3957: 3953: 3948: 3945: 3941: 3937: 3933: 3929: 3925: 3924: 3918: 3917: 3910: 3906: 3902: 3898: 3894: 3889: 3888: 3887: 3886: 3883: 3880: 3876: 3871: 3866: 3861: 3860: 3859: 3858: 3855: 3852: 3848: 3842: 3840: 3838: 3832: 3828: 3824: 3820: 3816: 3808: 3802: 3798: 3794: 3790: 3789: 3788: 3787: 3784: 3780: 3776: 3772: 3753: 3748: 3744: 3738: 3733: 3729: 3722: 3713: 3684: 3681: 3676: 3671: 3667: 3651: 3647: 3634: 3629: 3625: 3609: 3605: 3592: 3587: 3583: 3575: 3574: 3572: 3569: 3568: 3567: 3566: 3562: 3558: 3554: 3546: 3544: 3543: 3539: 3535: 3531: 3527: 3520: 3505: 3502: 3479: 3472: 3471: 3470: 3469: 3467: 3460: 3458: 3453: 3451: 3443: 3439: 3435: 3431: 3427: 3423: 3420: 3419: 3415: 3413: 3406: 3402: 3398: 3393: 3390: 3388: 3383: 3374: 3372: 3371: 3367: 3363: 3358: 3356: 3352: 3348: 3344: 3339: 3337: 3329: 3325: 3321: 3317: 3306: 3302: 3301: 3300: 3297: 3293: 3292:Shigeo Sasaki 3289: 3288:Marcel Berger 3285: 3277: 3275: 3274: 3270: 3266: 3258: 3254: 3251: 3247: 3244: 3240: 3237: 3233: 3230: 3226: 3223: 3219: 3216: 3212: 3211: 3210: 3208: 3201: 3197: 3190: 3186: 3182: 3178: 3172: 3168: 3164: 3160: 3157: 3153: 3152: 3151: 3150: 3147: 3143: 3139: 3134: 3133: 3132: 3131: 3127: 3123: 3114: 3112: 3111: 3106: 3102: 3098: 3079: 3075: 3049: 3041: 3037: 3033: 3028: 3024: 3016: 3012: 3008: 3005: 3000: 2996: 2986: 2982: 2978: 2975: 2970: 2966: 2958: 2954: 2950: 2945: 2941: 2934: 2929: 2926: 2919: 2918: 2917: 2916: 2895: 2891: 2887: 2882: 2878: 2874: 2869: 2865: 2861: 2856: 2852: 2837: 2833: 2823: 2819: 2815: 2811: 2807: 2806: 2805: 2804: 2803: 2802: 2801: 2800: 2793: 2789: 2785: 2780: 2779: 2778: 2777: 2776: 2775: 2767: 2763: 2759: 2758: 2757: 2756: 2755: 2754: 2746: 2745: 2744: 2743: 2742: 2741: 2736: 2732: 2728: 2724: 2720: 2719: 2718: 2717: 2710: 2709: 2708: 2707: 2701: 2700: 2699: 2698: 2695: 2691: 2687: 2683: 2682: 2678: 2673: 2672: 2668: 2667: 2666: 2665: 2661: 2657: 2652: 2650: 2645: 2640: 2634:=(1+Γ)/(1-Γ). 2633: 2628: 2624: 2615: 2608: 2604: 2596: 2593: 2586: 2583: 2579: 2575: 2572:An impedance 2570: 2563: 2559: 2555: 2551: 2547: 2543: 2539: 2535: 2532: 2531: 2530: 2529: 2525: 2521: 2516: 2514: 2509: 2500: 2498: 2497: 2493: 2489: 2484: 2481: 2475: 2473: 2472: 2468: 2464: 2455: 2453: 2452: 2448: 2444: 2435: 2431: 2427: 2419: 2413: 2412: 2411: 2410: 2406: 2402: 2393: 2391: 2390: 2386: 2382: 2374: 2370: 2366: 2365: 2364: 2363: 2359: 2355: 2351: 2342: 2338: 2330: 2323: 2322: 2321: 2319: 2315: 2311: 2307: 2299: 2296: 2290: 2288: 2287: 2284: 2283:MotherFunctor 2275: 2273: 2272: 2269: 2264: 2263: 2260: 2255: 2249: 2247: 2246: 2242: 2238: 2231: 2223: 2221: 2220: 2217: 2206: 2204: 2201: 2197: 2193: 2189: 2181: 2177: 2170: 2168: 2167: 2164: 2160: 2156: 2152: 2148: 2144: 2140: 2132: 2128: 2125: 2121: 2120: 2119: 2118: 2115: 2111: 2108: 2104: 2102: 2097: 2094: 2091: 2088: 2082: 2074: 2070: 2066: 2065:Adam majewski 2062: 2059: 2058: 2057: 2053: 2049: 2045: 2044: 2043: 2039: 2035: 2034:Adam majewski 2031: 2028: 2027: 2026: 2025: 2022: 2018: 2014: 1994: 1991: 1988: 1985: 1982: 1976: 1970: 1967: 1964: 1961: 1958: 1954: 1949: 1945: 1942: 1938: 1932: 1929: 1926: 1923: 1920: 1916: 1911: 1901: 1894: 1893: 1892: 1890: 1886: 1881: 1879: 1860: 1856: 1852: 1845:is mapped to 1829: 1826: 1818: 1797:is mapped to 1784: 1780: 1776: 1773: 1770: 1767: 1759: 1758: 1757: 1754: 1751: 1748: 1745: 1739: 1731: 1728: 1724: 1720: 1719: 1718: 1717: 1716: 1715: 1712: 1709: 1705: 1689: 1683: 1680: 1673: 1670: 1667: 1664: 1659: 1655: 1649: 1646: 1643: 1633: 1632: 1630: 1629: 1624:a</mi: --> 1620:2</mn: --> 1618:<mrow: --> 1614:c</mi: --> 1610:a</mi: --> 1606:4</mn: --> 1604:<mrow: --> 1603:-</mo: --> 1600:2</mn: --> 1598:b</mi: --> 1596:<msup: --> 1595:<sqrt: --> 1594:±</mo: --> 1591:b</mi: --> 1589:-</mo: --> 1587:<mrow: --> 1586:<mrow: --> 1579: 1575: 1571: 1570: 1560: 1544: 1541: 1536: 1533: 1530: 1524: 1516: 1512: 1504: 1503: 1499: 1485: 1478: 1474: 1469: 1466: 1463: 1460: 1457: 1451: 1448: 1442: 1434: 1430: 1422: 1421: 1417: 1401: 1398: 1393: 1387: 1379: 1375: 1367: 1366: 1362: 1346: 1343: 1338: 1335: 1332: 1326: 1318: 1314: 1306: 1305: 1302: 1301: 1300: 1299: 1295: 1294: 1293: 1292: 1289: 1285: 1282: 1276: 1270: 1265: 1261: 1257: 1256: 1255: 1254: 1251: 1243: 1241: 1239: 1235: 1231: 1227: 1216: 1212: 1208: 1204: 1203: 1202: 1201: 1200: 1199: 1196: 1175: 1169: 1164: 1157: 1152: 1146: 1137: 1136: 1135: 1128: 1126: 1119: 1117: 1110: 1108: 1105: 1099: 1092: 1090: 1086: 1082: 1075: 1070: 1066: 1062: 1046: 1043: 1038: 1028: 1020: 1015: 1005: 995: 983: 977: 974: 965: 957: 954: 951: 948: 939: 932: 931: 929: 928: 924: 920: 916: 915: 914: 897: 889: 884: 881: 876: 873: 870: 861: 852: 850: 831: 823: 818: 815: 810: 807: 804: 795: 772: 769: 766: 763: 758: 755: 752: 749: 740: 731: 724: 717: 712: 708: 707: 706: 705: 701: 696: 692: 691:modular forms 688: 683: 682: 681: 679: 676: 670: 669: 665: 662: 658: 654: 648: 646: 643: 637: 634: 625: 624: 620: 616: 612: 608: 604: 600: 599: 595: 591: 587: 586: 585: 583: 582:Lorentz group 575: 574: 571: 565: 564: 563: 559: 555: 551: 548: 545: 544:Lorentz group 538:Lorentz group 537: 535: 533: 525: 523: 522: 518: 514: 509: 506: 500: 496: 492: 488: 484: 483: 482: 481: 478: 477:65.183.252.58 465: 462: 459: 455: 454: 453: 452: 451: 450: 445: 442: 438: 417: 412: 408: 400: 397: 396: 395: 394: 390: 387: 386: 382: 378: 374: 370: 366: 365: 364: 362: 357: 355: 347: 345: 342: 339: 337: 333: 330: 328: 322: 321: 305: 301: 297: 292: 288: 265: 261: 257: 252: 248: 237: 234: 233: 226: 222: 221: 220: 214: 210: 206: 202: 195: 194: 193: 191: 187: 183: 179: 171: 167: 163: 157: 142: 138: 137:High-priority 132: 129: 128: 125: 108: 104: 100: 99: 91: 85: 80: 78: 75: 71: 70: 66: 62:High‑priority 60: 57: 54: 50: 45: 41: 35: 27: 23: 18: 17: 4531:Our article 4495: 4491: 4488: 4484: 4305:Möbius plane 4284:other fields 4280:Möbius plane 4232: 4175: 4171: 4158: 4154:zero divisor 4152: 4146: 4139: 4134: 3985: 3951: 3943: 3939: 3935: 3931: 3927: 3921: 3914: 3908: 3904: 3900: 3896: 3892: 3874: 3869: 3843: 3836: 3826: 3823:dual numbers 3814: 3812: 3775:Adumbrativus 3550: 3529: 3525: 3523: 3518: 3463: 3461: 3456: 3454: 3449: 3447: 3411: 3378: 3359: 3354: 3350: 3346: 3342: 3340: 3335: 3333: 3296:Kentaro Yano 3283: 3281: 3265:50.53.43.172 3262: 3204: 3175:— Preceding 3170: 3166: 3162: 3155: 3118: 3066: 2840: 2765: 2761: 2676: 2653: 2648: 2641: 2638: 2631: 2626: 2613: 2606: 2602: 2594: 2584: 2546:50.53.43.172 2537: 2517: 2504: 2485: 2482: 2479: 2459: 2439: 2417:siℓℓy rabbit 2397: 2378: 2345: 2328:siℓℓy rabbit 2300: 2297: 2294: 2279: 2265: 2256: 2253: 2227: 2210: 2182: 2178: 2174: 2171:Organisation 2158: 2154: 2150: 2146: 2142: 2138: 2136: 2133:quaternions? 2112: 2109: 2105: 2098: 2095: 2092: 2089: 2086: 2019: 2015: 2012: 1888: 1884: 1882: 1878:real numbers 1875: 1755: 1752: 1749: 1746: 1743: 1286: 1283: 1280: 1274: 1247: 1230:86.25.180.89 1221: 1192: 1132: 1123: 1114: 1103: 1100: 1096: 1087: 1083: 1079: 1064: 922: 918: 853: 732: 728: 698:chunks. -- 686: 671: 649: 638: 632: 630: 618: 614: 610: 606: 602: 593: 589: 578: 566: 560: 556: 552: 549: 541: 529: 503: 473: 436: 358: 351: 343: 340: 334: 331: 323: 239: 238: 235: 231: 230: 225:Evil saltine 218: 176:— Preceding 172: 168: 164: 161: 136: 96: 40:WikiProjects 4498:Hamishtodd1 2834:Comment on 2810:Smith chart 2644:Smith chart 2621:There is a 2574:Smith chart 2562:Smith chart 2560:Picture of 2542:p. 51, 3.19 2304:—Preceding 1623:<mi: --> 1621:<mo: --> 1619:<mn: --> 1613:<mi: --> 1611:<mo: --> 1609:<mi: --> 1607:<mo: --> 1605:<mn: --> 1602:<mo: --> 1599:<mn: --> 1597:<mi: --> 1593:<mo: --> 1590:<mi: --> 1588:<mo: --> 1224:—Preceding 1207:85.1.13.169 590:similitudes 182:96.55.28.32 112:Mathematics 103:mathematics 59:Mathematics 4564:Categories 3956:homography 3397:Salix alba 2508:involution 2268:Randomblue 2259:Randomblue 1819:the point 1760:the point 1107:language. 607:degenerate 4549:jacobolus 4521:jacobolus 4441:jacobolus 4403:jacobolus 4313:jacobolus 4307:is about 4290:, though 4265:jacobolus 4194:jacobolus 4160:zero ring 4123:jacobolus 4107:jacobolus 3851:jacobolus 3097:Freeboson 2592:impedance 2153:), where 1264:Hermitian 1250:Dysprosia 1195:John Baez 917:Usually, 849:Mathworld 3305:Sławomir 3177:unsigned 3105:contribs 2520:D.Lazard 2381:Sam nead 2369:Billlion 2354:Billlion 2306:unsigned 2281:sphere. 2188:unsigned 2048:D.Lazard 1226:unsigned 633:unlinked 361:Billlion 178:unsigned 4454:Quondum 4420:Quondum 4325:Quondum 4296:Quondum 4237:Quondum 4180:Quondum 4165:WT:MATH 4148:divisor 3969:Quondum 3879:Quondum 3847:Quondum 3430:Marvoir 3316:YohanN7 3165:, this 2766:mention 2488:Nielius 2443:Paxinum 2114:Xah Lee 2021:Xah Lee 1727:Xah Lee 1578:blahtex 1288:Xah Lee 1244:Fraktur 1069:Fropuff 627:group). 139:on the 30:B-class 4428:pages. 4224:found. 3942:, and 3094:etc.? 2401:Alunmw 2228:isn't 2216:Wrayal 2163:MarSch 2124:MarSch 678:(talk) 664:(talk) 645:(talk) 611:vector 603:vector 513:Joriki 215:Applet 36:scale. 4435:from 4250:Kusma 4211:Kusma 3156:group 2814:Steve 2762:about 2727:Steve 2656:Steve 2463:ΨΦorg 1708:KSmrq 1574:texvc 1284:thx. 716:linas 700:linas 4543:and 4502:talk 4397:etc. 4286:and 4254:talk 4215:talk 3821:and 3797:talk 3779:talk 3561:talk 3538:talk 3524:And 3457:true 3434:talk 3401:talk 3366:talk 3362:Daqu 3347:then 3320:talk 3294:and 3269:talk 3185:talk 3167:does 3142:talk 3138:Daqu 3126:talk 3101:talk 2818:talk 2788:talk 2784:Daqu 2782:etc. 2731:talk 2690:talk 2686:Daqu 2660:talk 2642:The 2550:talk 2524:talk 2492:talk 2467:talk 2447:talk 2424:talk 2405:talk 2385:talk 2373:talk 2358:talk 2335:talk 2314:talk 2254:Hi, 2230:this 2196:talk 2069:talk 2052:talk 2038:talk 2032:. -- 1234:talk 1211:talk 1089:2). 659:.--- 615:line 594:line 526:more 517:talk 495:talk 280:and 205:talk 186:talk 131:High 4552:(t) 4524:(t) 4444:(t) 4406:(t) 4316:(t) 4268:(t) 4248:. — 4197:(t) 4192:. – 4167:?). 4141:rng 4126:(t) 4110:(t) 3895:↦ ( 3872:and 3854:(t) 3793:Kri 3557:Kri 3450:not 3355:not 3336:one 3173:. 2580:of 2236:zro 2213:--> 693:or 675:CH 673:--- 668:CH 661:CH 655:of 642:CH 640:--- 491:C S 4566:: 4504:) 4350:↦ 4256:) 4217:) 4157:, 4151:, 4079:− 4038:↦ 3938:, 3934:, 3930:, 3907:+ 3905:cx 3903:)( 3899:+ 3897:ax 3799:) 3781:) 3726:↦ 3685:⋱ 3563:) 3540:) 3526:No 3506:2. 3494:^ 3480:χ 3436:) 3428:. 3403:) 3368:) 3343:if 3322:) 3314:. 3271:) 3263:-- 3187:) 3144:) 3128:) 3103:| 3080:μ 3076:σ 3034:− 3006:− 2820:) 2790:) 2733:) 2692:) 2677:if 2662:) 2552:) 2526:) 2494:) 2469:) 2449:) 2428:) 2407:) 2387:) 2360:) 2339:) 2316:) 2198:) 2071:) 2063:-- 2054:) 2040:) 1983:− 1965:− 1946:− 1927:− 1905:↦ 1889:bc 1887:− 1885:ad 1833:∞ 1805:∞ 1774:− 1665:− 1650:± 1644:− 1531::= 1464:− 1452:− 1449::= 1394::= 1333::= 1236:) 1213:) 1021:− 987:¯ 969:¯ 943:↦ 893:¯ 885:− 874:− 865:↦ 851:. 827:¯ 819:− 808:− 799:↦ 744:↦ 519:) 497:) 439:. 422:∞ 418:∪ 356:. 298:− 258:− 207:) 188:) 4500:( 4452:— 4385:, 4379:d 4376:+ 4373:z 4370:c 4365:b 4362:+ 4359:z 4356:a 4347:z 4252:( 4213:( 4121:– 4082:1 4075:) 4071:d 4068:+ 4065:x 4062:c 4059:( 4056:) 4053:b 4050:+ 4047:x 4044:a 4041:( 4035:x 4013:n 4008:R 3967:— 3959:" 3944:z 3940:d 3936:c 3932:b 3928:a 3911:) 3909:d 3901:b 3893:x 3875:C 3870:R 3839:" 3837:R 3795:( 3777:( 3754:z 3749:k 3745:b 3739:+ 3734:k 3730:a 3723:z 3677:+ 3672:2 3668:a 3652:1 3648:b 3635:+ 3630:1 3626:a 3610:0 3606:b 3593:+ 3588:0 3584:a 3559:( 3536:( 3519:" 3503:= 3500:) 3490:C 3483:( 3462:" 3432:( 3399:( 3364:( 3318:( 3284:n 3267:( 3183:( 3140:( 3124:( 3107:} 3099:{ 3050:] 3042:3 3038:x 3029:0 3025:x 3017:2 3013:x 3009:i 3001:1 2997:x 2987:2 2983:x 2979:i 2976:+ 2971:1 2967:x 2959:3 2955:x 2951:+ 2946:0 2942:x 2935:[ 2930:= 2927:X 2901:) 2896:3 2892:x 2888:, 2883:2 2879:x 2875:, 2870:1 2866:x 2862:, 2857:0 2853:x 2849:( 2816:( 2786:( 2729:( 2688:( 2675:( 2658:( 2649:z 2632:z 2627:z 2619:. 2617:0 2614:Z 2612:/ 2610:L 2607:Z 2605:= 2603:z 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