Talk:Riemann sphere

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[edit] This article needs lots of work

Take a look at the short paragraph below:

[edit] Complex structure

The complex manifold structure on the Riemann sphere is specified by an atlas with two charts and coordinates z and w

z:\hat{\mathbb{C}}\setminus\{\infty\} \to \mathbb{C}\,
w:\hat{\mathbb{C}}\setminus\{0\} \to \mathbb{C}\,

The transition function between the two patches is w = 1/z, which is clearly holomorphic and so defines a complex structure. To see that these charts give the topology of the sphere note that we can give an atlas on S2 by stereographic projection onto the complex planes tangent to the north and south poles respectively. Labeling points in S2 by (x1, x2, x3) where x_1^2 + x_2^2 + x_3^2 = 1, we have

z = \frac{x_1+i x_2}{1+x_3}
w = \frac{x_1-i x_2}{1-x_3}

which satisfies the equation w = 1/z. In terms of standard spherical coordinates (θ, φ)

z = e^{i\phi}\tan\frac{\theta}{2}
w = e^{-i\phi}\cot\frac{\theta}{2}

[edit] Problems

  • For the map:
z:\hat{\mathbb{C}}\setminus\{\infty\} \to \mathbb{C}\,,
what is the formula?
  • For the map :w:\hat{\mathbb{C}}\setminus\{0\} \to \mathbb{C}\,,
what is the formula?
  • The equations
z = \frac{x_1+i x_2}{1+x_3}
w = \frac{x_1-i x_2}{1-x_3}

are not in the proper context, they implicitely assume that the connection between the Riemann sphere and the normal sphere is already established, while it was not.

Also, in a paragraph below in text one says:

When the sphere is given the round metric...

Same problem. This makes no sense until a correspondence between the normal sphere and the Riemann sphere is explicitely stated, and that correspondence is fixed, so there is no ambiguity. Oleg Alexandrov 01:57, 15 Feb 2005 (UTC)

Perhaps the wording in the article is not the best. If you think things would be clearer, feel free to make it more rigorous. The idea is very simple though. You have two coordinate charts (both copies of C) with coordinates z and w. The transition function on the overlap (where both z and w are nonzero) is given by z = 1/w. That's all the information needed to specify the complex manifold. The equations relating z and w to the x's is defining the relationship between the geometrical unit 2-sphere and the Riemann sphere. -- Fropuff 04:23, 2005 Feb 15 (UTC)

From my knowlege of differential geometry, to define a manyfold you need (a) some charts covering it (b) a map from each chart to a subset of R^n (C^n) (c) the transition function (that follows from (b) actually). You did (a) and (c), but did not specify (b). I will do some changes in this article tomorrow. Oleg Alexandrov 04:40, 15 Feb 2005 (UTC)
My comments were exaggerated, sorry. One could have realized that by z you mean the identity map, z→z, and that by
z = \frac{x_1+i x_2}{1+x_3}
you mean the map which starts at a point (x_1, x_2, x_3), takes as output this fraction, which you denote by z, and which you further identify with the chart going from \mathbb{C} to \hat{\mathbb{C}} or the other way around. But making these explicit as I did, certainly does not hurt. Oleg Alexandrov 20:04, 15 Feb 2005 (UTC)

No worries. It is a little confusing with C serving as both part of the manifold and as coordinate charts. One can be clever with the notation but it is perhaps more confusing in the end. -- Fropuff 02:30, 2005 Feb 16 (UTC)

One certainly should not make notation stay in the way of understanding. Point taken. The only reasons I introduced the notation f and g was to make explicit the charts, as unless you work all your life in complex analysis and know that z is the default variable, one would not understand how z is defined. If you feel notation is getting in the way somewhere, please let me know, I don't quite see it myself.
You said that it is a little confusing with C serving as both part of the manifold and as coordinate charts. I don't know what we can do about it. After all, that's how the Riemann sphere is defined, as
\hat{\mathbb{C}} = \mathbb{C}\cup\{\infty\}.
Or this is not what you meant? Oleg Alexandrov 02:51, 16 Feb 2005 (UTC)

That is what I meant. But as you say, there is little that can be done about it. I think what is there now works fine. -- Fropuff 03:01, 2005 Feb 16 (UTC)

[edit] Complexity of the article (presentation angle)

(No puns intended on complexity or angle)

I have found several much simpler explanations for the complex sphere. Examples include the one in Penrose's The Road to Reality or even the more technical one in Needham's Visual Complex Analysis. Their common simplifying factor is presenting it in terms of geometry, with diagrams to boot.

Presenting the Riemann sphere in terms of a set theory and mappings seems unnecessarily abstract. Giving the transformations and set-theoretic definitions, IMO, should be secondary to giving the geometric derivations of mapping the complex plane (+ infinity) to points on a sphere.

Disclaimer: I am a high school senior learning these things on my own. I don't know how textbooks do it. I just look at what I'm given and judge. In the case of Penrose and Needham, I say "this makes sense and is really cool!" But for Wikipedia, I come out of it more like "huh? Why do it this way? What the heck does this even have to do with spheres?"

So basically, I don't pretend to be an authority on this. —The preceding unsigned comment was added by DomenicDenicola (talk • contribs).

Well, if you wish, you could start by adding diagram and some text, and we may help. What you say makes sense. Oleg Alexandrov (talk) 21:55, 8 December 2005 (UTC)
The presentation in the article is geometrical; at least to a graduate student in mathematics. But, you are right, the Riemann sphere is often encountered by bright high school students long before graduate school. This page could do with some pictures and a gentler introduction. -- Fropuff 22:07, 8 December 2005 (UTC)

[edit] Working draft of geometric rewrite

(Notes: this is meant to be standalone, so when it's done we would find overlap from other sections and remove it here or there.)

Define \widehat{\mathbb{C}} = \mathbb{C}\cup\{\infty\} (i.e. the complex numbers joined with the point at infinity). The Riemann sphere is based on the transformation from \widehat{\mathbb{C}} to \widehat{\mathbb{C}} and is in the form

w = f(z) = \frac{1}{z},

where w, z \in \widehat{\mathbb{C}} and \frac{1}{0} = \infty.

We visualize the Riemman sphere as a sphere in 3-space, i.e. in \mathbb{R}^3. Every point on the sphere has both a z value and w value, related by the above transformation. That is, f(z) transforms the sphere onto itself.

[edit] Stereographic projection

To establish the correspondence between points in the extended complex plane and the Riemann sphere, we first place the z plane across the sphere's equator. We then use stereographic projection from the south pole of the sphere. This is done by drawing a line from the south pole that intersects both the sphere and the complex plane; a unique, one-to-one correspondence is then established between points on the complex plane and points on the Riemann sphere. Note that points on the complex plane inside the unit circle will map to the upper hemisphere, and points outside will map to the lower hemisphere.

In order to complete this one-to-one correspondence for the extended complex plane, we define the south pole to be z = \infty. Note that the north pole is z = 0.

The correspondence between the w plane and the Riemann sphere is done in much the same way, simply "upside down." That is, the w plane is an equitorial plane oriented oppositely to the z plane, such that w = 1,i, − 1, − i matches to z = 1, − i, − 1,i. We then perform the stereographic projection from the north pole, and similarly define the north pole to be w = \infty. Now, every point on the sphere has both a z and w coordinate, related by the transformation above.

[edit] Geometric features of note

The equator of the sphere is the unit circle in the complex plane; in a similar fashion, circles can be found for the imaginary line and real lines. Note that these are shared between the two projections, because the relation w = \frac{1}{z} is holomorphic.

This is a specific case of how stereographic projection maps all lines and circles in the complex plane to circles on the Riemann sphere. The reason that lines are mapped to circles is that a line with infinite length can simply be thought of as a circle that passes through the point at infinity.

[edit] Möbius transformations

Möbius transformations, which send \widehat{\mathbb{C}} to \widehat{\mathbb{C}}, are often visualized as acting on the Riemann sphere. They are in the form

t = f(z) = \frac{az + b}{cz + d},

where t, z \in \widehat{\mathbb{C}}, a, b, c, d \in \mathbb{C}, and ad - bc \neq 0. They map the sphere to itself, preserving important features such as angles and circles/lines. This is because they are only composed of dilations, translations and inversions.

[edit] Problems

I think I've confused myself with the whole Möbius transformation thing. I believe my explanation is only valid for the transformation w = 1 / z. So how do Möbius transformations fit in? I know they map Riemann spheres to Riemann spheres, but I don't think they do so in the manner I described (simple projection through the matched planes).

I think I fixed this, although I'm a little shaky on the need for/accuracy of the new Möbius transformation section. I just thought that since they were so important to the Riemann sphere the article should say something about them. Domenic Denicola 00:20, 10 December 2005 (UTC)

Does this sentence even make sense? I might not be saying what I'm trying to say... "Note that these are shared between the two projections, because the Möbius transformation is holomorphic."

Made irrelevant by the above fix; I certainly know that w = \frac{1}{z} is holomorphic! Domenic Denicola 00:20, 10 December 2005 (UTC)


Comments on explanation? I really like it, but hey, that's because it makes sense to me.

I really need diagrams, especially ones just showing the sphere superimposed with at least the z plane and with 1, − 1,i, − i labelled.

This will depend on how we place the planes, but I could scan Penrose if we stay with equitorial planes. Domenic Denicola 00:20, 10 December 2005 (UTC)

Domenic Denicola 19:50, 9 December 2005 (UTC)

Scanning Penrose is bound to violate copyright laws. I really think we should stick with tangent planes anyway. You could put up a request on Wikipedia:Requested_pictures#Mathematics. -- Fropuff 00:28, 10 December 2005 (UTC)
OK; I'll do some searching around and then probably put in a request. I think I found why Penrose likes the equatorial version; it works better for Twistor theory. See this page with diagrams.

[edit] Comments of above suggestion

  • Mobius transformations are usually defined in terms of the Riemann sphere and not the other way around. One should leave them out of the definition to avoid circular logic. The only transformation you have to worry about is w = 1/z.
I'm a litte confused as to what exactly the point of having a w coordinate even is then. What does it give you? Why not just map a z plane onto a sphere?
(Speculation) Maybe it's because Riemann surfaces are always defined in terms of transformations or functions? I don't have much background on those either, but e.g. I know that w = log(z) has a twisting spiral plane surface; maybe you need the transformation w = 1 / z to get a sphere surface. Domenic Denicola 00:32, 10 December 2005 (UTC)
  • It is more common to stereographically project onto tangent planes rather than to planes through the equator. This has the added advantage of making the z-plane and the w-plane geometrically distinct.
After thinking about this statement for a while, I realized you meant planes tangent to the poles (right?). I guess it makes sense, but (probably just because I read the stacked planes version first) I don't like it as much. Could you find a web page that shows this in more detail, so I can hopefully be enlightened on why it's a nice way of doing things? I think we should change it too, if it's the conventional way of doing it; however, I'd like to understand the new version more thoroughly first. Domenic Denicola 00:32, 10 December 2005 (UTC)
  • The section on Complex structure in the article needs to remain as it shows how the Riemann sphere takes on the structure of a 1-dimensional complex manifold. I would suggest adding your material (sufficiently revised) as an introductory/motivation section.

-- Fropuff 21:15, 9 December 2005 (UTC)

Agree with Fropuff. Now, the current article introduction, saying
In mathematics, the Riemann sphere, named after Bernhard Riemann, is the unique simply-connected, compact, Riemann surface.
is intimidating indeed. I would start the article by saying the Riemann sphere is obtained by imagining that all the rays emanating from the origin of the complex plane eventually meet again at a point called the infinity, in the same way that all the meridians from the north pole of a sphere get to meet each other at the south pole. I don't know. Ideas? Oleg Alexandrov (talk) 22:08, 9 December 2005 (UTC)
Yes, it would be good to have something like that as the introduction. Another good blurb is "a useful visualization of the extended complex plane, especially when doing Mobius transformations." Other ideas would be good too :). Domenic Denicola 00:32, 10 December 2005 (UTC)
Feel free to add this kind of stuff, looks good. We may edit later if need be. About pictures. I could make a picture if necessary, but please note that we already have a picture (the very top of the article), and putting a sphere and a plane in the same picture while showing how the correspondance goes might not be possible. Oleg Alexandrov (talk) 02:03, 10 December 2005 (UTC)
I think these are some good samples of what I am looking for: sample 1 sample 2 sample 3.
Also, I am beginning to wonder about Fropuff's statement that tangent to the poles is more common. At least in stereographic projection---if not in Riemann spheres, necessarily---it seems the equitorial approach crops up a lot more often in my (highly unscientific) Google image search.

[edit] Post-rewrite comments

I deleted these sentences because I wasn't sure they worked for tangent planes:

Note that points on the complex plane inside the unit circle will map to the upper hemisphere, and points outside will map to the lower hemisphere.
The equator of the sphere is the unit circle in the complex plane; in a similar fashion, circles can be found for the imaginary line and real lines. Note that these are shared between the two projections, because the relation w = \frac{1}{z} is holomorphic.

If they only fit in with the equitorial projection, I could add them in the section on that; alternatively, if they work in both, I'll restore them to the appropriate places. I need guidance, however.

The planes are still switched in orientation, right?

Hope everyone likes the results!

Domenic Denicola 06:32, 13 December 2005 (UTC)

The rewrite looks good, thanks. I have some comments though.
1. I don't understand the sentence
The Riemann sphere is based on the transformation from ::::\widehat{\mathbb{C}} to \widehat{\mathbb{C}} in the form
w = f(z) = \frac{1}{z},
where w, z \in \widehat{\mathbb{C}} and \frac{1}{0} = \infty.
and I don't think it belongs in the elementary introduction. The article disucsses Mobius transformations later.
I didn't mean this as a Mobius transformation. Perhaps it is not necessary, but I think since we talk a lot about how w and z coordinates are related it should be there. I was under the impression that all Riemann surfaces were representations of some function (e.g. w = log(z) is a spiraling plane), and the Riemann sphere was the representation of w = 1 / z. The appropriate way to introduce this to someone not familiar with Riemann surfaces (in order to keep this article free from "required reading") seems to say that it is "based on the transformation." Does that sound right? How could I phrase it to make it more understandable? Domenic Denicola 20:49, 13 December 2005 (UTC)
2. The sentence: "To establish the correspondence between points in the extended complex plane and the Riemann sphere," is not right. The Riemann sphere and the extended complex plane are the same thing. The correspondence is between the extended complex plane (also known as the Riemann sphere) and the ususual sphere in R^3, a correspondence which justifies the name.
Proposed new section (please check for accuracy):
To establish the correspondence between points in the extended complex plane and a 3-sphere, we first place the z plane tangent to the sphere's north pole. We then use stereographic projection from the south pole of the sphere. This is done by drawing a line from the south pole that intersects both the sphere and the complex plane; a unique, one-to-one correspondence is then established between points on the complex plane and points on the 3-sphere. The "Riemann sphere" is thus a name for either of these structures, as they are mathematically equivalent. Domenic Denicola 20:49, 13 December 2005 (UTC)
I don't know. The point is that the Riemann sphere is not a sphere to start with, it is just the complex plane plus the point of infinity. That's be definition. Now, one can explain why it is called a Riemann sphere, but it does not chang ee the fact. Oleg Alexandrov (talk) 21:27, 13 December 2005 (UTC)
3. I am not sure I like the way the article is split in sections and subsections.
Overall, looks good, thanks, great job! These are all minor things. Oleg Alexandrov (talk) 16:09, 13 December 2005 (UTC)
Did Fropuff fix that for us? And thanks :). Domenic Denicola 20:49, 13 December 2005 (UTC)

[edit] Baffled

This article leaves me utterly confused. I don't expect to grasp the formulas toward the bottom of the page in any math-related article; I don't have sufficient background. But none of this makes any sense at all, not from the first few words.

How is a Riemann sphere distinct from an ordinary sphere? Explain it so my daughter can understand it. She knows the difference between a sphere and a circle; is this the same thing? Circle, sphere, Riemann sphere? (Square, cube, hypercube.)

Is there any difference between Riemann sphere and Riemann space? If so, what? Say it without any numbers or special words. If I lived on the surface of a Riemann sphere, what would be different about my life? Would all my doughnuts turn into coffeecups? Are squares still square?

Is there more than one Riemann sphere? Can they come in different sizes? (I suspect the answers are no and no.) Could I even tell if I did live in a Riemann sphere? It looks to me, offhand, as if the thing is of infinite size. Wouldn't any finite zone or section (my local known universe) always appear Euclidean? Is any point on the Riemann sphere distinguished? If so, what would happen if I stood there? Would I blow up? Would my left and right hands disappear or get stuck together? Could I even tell?

What is the use of this thing? How can I apply it and to what? Automotive engineering? Faster-than-light spacecraft theory? Molecular biology? Game theory? Can I win a bar bet with it; if so, what's the bet?

I came here because graphics were requested; I'm willing to do graphics. But first, somebody will have to tell me what it is. John Reid 20:14, 14 April 2006 (UTC)

Good points, but you are expecting too much from an article on an abstract topological concept. So here we are:
The Riemann sphere is not a sphere, it is the complex plane plus an extra point (paragraph 1)
The Riemann sphere is topologically equivalent to the real sphere (paragraph 2). Sorry, but that topological homeomoprhisms can't be explained easily to people who don't know what topology is.
The question about different sizes of Riemman spheres does not make sense, again, a Riemann sphere is the complex plane with an extra point.
One of its uses is to conveniently deal with meromorphic functions, or otherwise, functions taking infinite values. I know that is not what you are looking for, but this is an abstract concept, used in other areas of math, and not directly in real life.
In short, your questions are appreciated, and you are right that there is always room for improvement for abstract math articles, but you should also not expect way too much, some things are abstract and complex because they must be so. Oleg Alexandrov (talk) 20:23, 14 April 2006 (UTC)

The fine points may be lost on a layman but please have the courtesy not to look down your nose at me. I have enough background in topology to understand why I can remove my vest without taking off my coat. I know that if I glue together all four corners of a checkerboard and glue (along each edge) four squares to four squares then I have created a bag-like thing which is not equivalent to a sphere; it is not even a manifold because there are distinguished points. I know why I can never comb a hairy ball. And I even have a pretty fair idea why the N-color problem has a different solution on a torus. If you can't explain the subject of this article to me, I think you don't have a clear grasp of the concept -- only what you have been told. No offense intended.

The question about different sizes makes a great deal of sense. One of the first things I do when trying to understand a new concept is to search for limits. Is there one only or more than one? Does it come in more than one size or variety? As I said, I suspect there is only one, of infinite size. You still have not told me if the point at infinity is distinguished. Indeed, you've failed to answer most of my questions. That's okay; but if you don't know the answers then please don't be so dismissive of the questions.

Nothing useful is purely abstract. Experts who work with a concept regularly may be comfortable with an abstract representation; but there is always some connection to reality -- otherwise the concept is nonsensical. You may have difficulty associating this topic to a practical aspect of life but I suggest that there is such a connection. If not, I'm tempted to say that however important it may be to a specialist, it holds no possible interest for the general reader. John Reid 22:20, 14 April 2006 (UTC)

I guess I found your question to be rather inflammatory, with wording like "utterly confused, baffled, etc". It was as if you are were saying "you people did an awfully bad job here" rather than "Hi all, I don't know a huge lot about topology, so would you be so kind to explain". But it seems that I answered in kind.
That said, I will take some time to think of a gentle answer to your question. Oleg Alexandrov (talk) 23:28, 14 April 2006 (UTC)
I am completely in favour of making all concepts as accessible to as many people as possible by describing them first in the most undemanding intuitive way possible, but it is wise to accept the limits of what is reasonable. It would be dishonest to claim that all mathematical topics can be made accessible to readers with all levels of knowledge. The first paragraph (if a reader ignores words they might not understand) should get the partial message across that the Riemann sphere is the complex plane plus one point called "infinity" which can be viewed as a 2-dimensional sphere (the stereographic projection section explains how this works). This is almost as much as is possible to get across without the concept of a complex manifold. The one thing I think is missing from the introduction is a mention of the fact that the point at infinity is used to extend the domain and the range of many complex functions in a useful way. Without a knowledge of complex analysis, the most honest thing to say would be that this would not be a useful article to read at all. Mathematics is hierarchical, and in many cases it is necessary to have a fair knowledge of certain subjects in order to get to grips with more advanced ones: I would not expect to understand an advanced text on any subject where I didn't understand the prerequisites, although I have often wished I could. This is simply being realistic. As an elementary analogy, in order to understand the purpose of the sieve of Eratosthenes (which I fondly remember executing by hand late in primary school) it is necessary to first study multiplication. Paraphrasing Euclid's honest words to king Ptolemy I, "there is no royal road to" differential "geometry" Elroch 01:03, 15 April 2006 (UTC)
Problems with the introduction at the moment are that there are a number of things mentioned which are really part-truths and distractions from the main idea. The Riemann sphere is not constructed using lines, nor is it constructed using the 1-point compactification of the plane. The first is a geometrical concept, the second a topological one. The Riemann sphere is actually constructed (as a Riemann surface) in one step using two charts, as explained in the article, and the topology derives incidentally from this. The message to get across is:
(1) that using the function 1/z identifies the extended complex plane with zero omitted to a second copy of the complex plane, and allows us to transfer the notions of differentiability we know about on the complex plane to the extended complex plane with zero omitted.
(2) fortunately, this notion of differentiability is identical with the usual one on the overlap with the complex plane of the space in (1) (i.e on the complex plane with zero omitted). Hence it gives us a well-defined notion of differentiability on the whole extended complex plane.
(3) this notion of differentiability on the whole extended complex plane allows us to talk about the differential properties of functions from the extended complex plane (or open subsets of it) to itself (essentially, complex functions that may also be evaluated at infinity and may take the value infinity). A key example is that meromorphic functions extend in a unique way.
(4) almost incidentally, the topology on the Riemann sphere that derives from the differential structure is that of a 2-dimensional sphere. Elroch 23:52, 15 April 2006 (UTC)

To reply to John Reid, from the top down, here it goes. The Riemann sphere is just the complex plane with an extra point added in, called the point at infinity. For analogy, look at the real line. There, when dealing with limits, it is convenient to pretend that there exist two points ∞ and -∞ which are endpoints of the real line. Then ∞+∞=∞, and all other formal rules makes it easier to deal with limits without worrying much about particular cases of infinite limit.

In the same way, one can pretend that all rays in the complex plane originating from 0 actually have an endpoint, and they all eventually meet at infinity, a point far-far away (not accurate as Elroch mentions above, but helpful in imagining things).

The Riemann sphere is not the same as the usual sphere, but they are topologically equivalent. Imagine a normal sphere, remove the north pole, and make the obtained hole there larger and larger (assume the sphere is made of very flexible rubber). Eventually, that sphere without a point can be flattened in a plane, the complex plane. The original north pole corresponds to the point at infinity in the complex plane.

There is only one Riemann sphere, as the point at infinity is just a symbol, its actual nature is not relevant. In the same way that there exists essentially one normal sphere. The radiuses may differ, but any sphere can be deformed gently into another sphere, without tearing the surface. In exactly the same way a sphere is the same as the surface of a cube, but not with the surfce of a donut.

You can't say if any portion of the Riemann sphere appears Euclidean, or whether it is infinite in size or not. That because there is no concept of distance and size on the Riemann sphere. Any portion of the sphere can be stretched/shrank in any way as long as the sphere does not burst or separate patches merge.

The Riemann sphere does not get applied directly much beyond math, or otherwise I never heard of it. It is a useful construct, but rather abstract.

I was under the impression that it was used in quantum theory as a way of visualizing CP1, which represents the spin state of an electron. At least, that is, by Roger Penrose. Domenic Denicola 03:46, 16 April 2006 (UTC)

Above I talked about the topology of the Riemann sphere, not its differential geometry . But that would be harder to explain.

I don't know how satisfactory you found the answers. Try to read them though, and let me know if you have questions. Oleg Alexandrov (talk) 02:37, 16 April 2006 (UTC)

Thank you. My understanding of this topic has been broadened. Here is what I have gleaned so far:
(1) This subject is not a geometric thing at all, but only topological. Size and distance are meaningless here, so there's no point discussing things like the sum of the angles of a triangle.
(2) The subject is a manifold; no point is distinguished and a bug at any point would perceive the same local universe as one at any other. Indeed it's just another genus-0 spheroid, topologically.
(3) This subject has no practical application. It is a pure abstraction of interest only to specialists. There are no concrete examples possible, no jokes, no bar bets, nothing that might reach beyond mathematics into the commonplace world.
(4) This subject is highly resistant to any treatment that might make it accessible or useful to a general audience. It doesn't have a hot or sexy handle sticking out of it.
Editors are welcome to enlarge on and correct this understanding but I'm content. It might be nice if there was an attempt, very early in the introduction, to somehow indicate to the general readership that the density of the article is intrinsic to the topic and not an artifact of poor or obscure writing. This is beyond my capabilities; I'm moving on. John Reid 14:46, 17 April 2006 (UTC)

[edit] Recent changes

Elroch made a lot of good edits to this article. It is now more mathematically correct, but it is hard to understand for somebody not knowing math however. I believe the geometric viewpoint, which, if not entirely accurate, was helpful in illustrating what is going on. Wonder what you think. Oleg Alexandrov (talk) 15:09, 17 April 2006 (UTC)

Thank you for your kind comment, Oleg. However, I now realise I have been unnecessarily narrow in my viewpoint. The thing that has shifted my viewpoint is finally realising (as Hipparchus did a little earlier :-) ) that the stereographic projection, although of course not preserving distances, is conformal. This gives the geometrical viewpoint a fully justified role (with all non-trivial holomorphic maps, including Möbius transformations) being conformal on the geometrical sphere). I wish this had sunk in before I made my second "improvement" to the introduction. Oh well, at least this isn't on paper. Of course this justifies increasing the prominence of the geometrical viewpoint again, which should help to address your suggestion. Elroch 19:11, 17 April 2006 (UTC)
I have now returned the geometrical viewpoint to the introduction, and added a little more detail. Hopefully this will mean the net change has been positive, and anything that is still less than ideal will be flagged or fixed by those more perceptive than me. Elroch 19:48, 17 April 2006 (UTC)
Thanks, your edits improved the article. Oleg Alexandrov (talk) 03:51, 18 April 2006 (UTC)
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