Talk:Inversive geometry/Archive 1

My reversions
I just reverted Patrick's recent changes. There are there goodies to keep maybe, but that can be figured out later.

Invertive geometry is an elementary geometry topic, acessible (and sometimes taught to) high school students. It is highly useful in sinthetic geometry for doing proofs. One can do invertive geometry knowing nothing about analytic geometry, all one needs to know is again, elementary geometry, what is a circle, line, what is a reflection, symmetry etc.

Starting this article with the full-blown generalization to n dimensions adds very little value to the aricle (if you are a mathematician, the generaliation is obvious). However, starting with the generalization greatly reduces its value for undergraduate students or for people who only know elementary geometry.

Please remember an important lesson. Keep articles accessible. One can read in the math style manual about that. This is a general purpose encyclopedia. Keep your math genius ego in check, and start an article at the most acessible place to the reader. If you got the reader hooked on, he/she might be willing to read on. If you start an article with "in n-dimensional space" the reader will stop here, unless the reader is at least as smart as you to start with. Oleg Alexandrov (talk) 17:20, 8 October 2005 (UTC)

Change to the introduction
I removed some stuff from the introduction. Introduction is meant to be a very simple description of what the article is about. Inversive geometry is about treating circles and lines the same, and tranformations which map these "generalized circles" to themselves. So why not just say that? Whether that eventually turns out to be conformal geometry, reflections and all that, is not that important to put it in the very first sentence. Oleg Alexandrov (talk) 13:03, 17 October 2005 (UTC)

QBAsic
Isn't the QBasic program a bit out of place? Hardly anyone uses QBasic anyway. Phys 05:31, 1 January 2005 (UTC)


 * I agree. It would be neat to have a link to a web page with a java program that would let someone do this but this takes up a lot of room and is a program besides, which doesn't seem to fit an encyclopedia. Gene Ward Smith 07:51, 3 May 2006 (UTC)


 * Agree too. I removed that text. The external link
 * Inversion: Reflection in a Circle at cut-the-knot
 * in the article seems to have a script at the other end illustrating inversions, that should be enough. Oleg Alexandrov (talk) 14:53, 3 May 2006 (UTC)

Anti-deSitter stuff removed
Someone might want to fix this, but it's hardly clear this is the best place for it in any case:

This is the Wick-rotated version of the AdS/CFT duality. In fact, since most calculations are performed in the Wick-rotated model, this is the duality which is really being used.

See also AdS/CFT. Gene Ward Smith 21:57, 4 May 2006 (UTC)

relation to mobius trans
here's a proposed gist of the content:

Circle inversion plays a critical role in Möbius transformation. A mobius transformation can be decomposed into a sequence of rotation, dilation, translation, and circle inversion. Of these, the circle inversions gives mobius transformation a critical characteristics. In particular, circle inversion is the only map among the composition that is a non-trivial conformal map.

Xah Lee 16:22, 15 July 2006 (UTC)

critical but missing topics: sph inv, mob trans, stereo proj
The following contents, about how sphere inversion, stereographic projection being a special application of sphere inversion, and how circle inversion is the gist of mobius transform, should be written. I started with the following in the article, but it got deleted thru political struggle. By the outcome of a diplomatic relation, a request is made that i put them here. Here they are:

Inversions in three dimensions
The 3-dimensional version of inversion is analogous to the 2-dimentional case.

The inversion of a vector $$P$$ in 3D with respect to a sphere centered on the origin with radius $$r$$ is a vector $$P'$$ such that $$|OP|\, |OP'|=r^2$$ and $$P'$$ is a positive multiple of $$P.$$

As with the 2D version, a sphere inverts to a sphere, except that if a sphere passes through O, then it inverts to a plane. Any plane not passing through 0, inverts to a sphere touching at O.

Stereographic projection is a special case of sphere inversion. Suppose, we have a stereographic projection with a sphere $$S$$ of radius 1 sitting on the origin $$O$$ of plane $$P$$, and the north pole $$N$$ being the projection point. Then, consider a sphere $$S_2$$ of radius 2 centered at $$N$$. The inversion in respect to $$S_2$$ transforms $$S$$ into its stereographic projection.

Xah Lee 16:09, 15 July 2006 (UTC)


 * hi Oleg. Thanks for the edit. This phrase: “P' is a positive multiple of P.” i didn't understand in the first reading. Perhaps we can say that “P' has the same direction as P”. This way, the phrasing gives it a more geometric interpretation inline with sphere inversion. Xah Lee 22:17, 17 July 2006 (UTC)

Too narrow
I'm concerned that this discussion under "inversion" considers only circle inversion. Inversion with respect to a point is also an important geometric transformation. --KSmrqT 19:56, 6 April 2007 (UTC)

Not mathematical rigour
"(a circle with infinite radius)", this strikes me as unmathematical. 'infinite radius' is not a defined term. What is meant is that any line is the limit of a circle whose radius may grow boundlessly. What many people seem to forget about the 'limit' is that it is strictly a value or state to which we can get as close as we desire by taking a variable as close to a constant as we want or as great or small as we want. The notations $$\lim_{h \to \infty} f(h) = L$$ and $$\lim_{h \to c} f(h) = \infty$$ are a bit unpleasantly chosen and nowhere mean that h is 'approaching infinity' or that the value of f(h) is 'approaching infinity'. Approaching infinity is a nonsensical term. Firstly, intuïtively, you will realize that you are getting nowhere closer to infinity as your grow bigger. Secondly. If we were to take $$d(h,\infty)$$ on $$\mathbb{R}$$ we would get simply $$|h - \infty|$$, given the commutative nature of subtraction under absolutes we may assert $$|\infty - h| = |\infty| = \infty$$ because of subtracting any real number from infinity remains infinity. So the distance between h and infinity will simply remain infinite. What is meant in the first limit is that h can grow boundlessly and in the second that f(h) can grow boundlessly. A particularly unhandy notation. Especially because it violates the traditional use of the equality sign which is extremely standard and using for something which is not the equality relation at all. It is not really transitive here now is it? Niarch (talk) 15:41, 23 April 2008 (UTC)


 * I don't see a problem. You are right that any formal definition would have to consider limits, but that shouldn't be too hard.  A circle of finite radius can be uniquely described by a point $$p$$, a tangent direction $$t$$ at that point, and a radius $$r$$.  For any circumferential distance $$d$$, consider the point $$p'(r,d)$$ a distance $$d$$ away from $$p$$ on a circle of radius $$r$$.  Then unless I'm mistaken, both $$\lim_{r\rightarrow\infty}\{p'(r,d):d\in R\}$$ and $$\{\lim_{r\rightarrow\infty}p'(r,d):d\in R\}$$ are the set of points on the line through $$p$$ with direction $$t$$.
 * Interpreting it rigorously is straightforward for those who care about maths, and unnecessary for those who don't. LachlanA (talk) 18:24, 26 May 2008 (UTC)

Split Inversion (geometry) to Circle inversion and Circular transformation
I propose to split Inversion (geometry) (the present page) to Circle inversion and Circular transformation.


 * Circle inversion would carry most of the present content of the page. Technically, this article would be Moved to its new title.
 * Circular transformation would discuss general circle-preserving maps, known variously as circular transformations and extended Möbius transformations. It would also discuss the groups of such transformations, variously known as the circular group, extended Möbius group, and inversive group. Via the Erlangen program, this article would also discuss inversive geometry.
 * Inversion (geometry) is a vague term and should be a dab page, pointing at minimum to Inversion (geometry) and Inversion in a point. (Wow, I think I just agreed with KSmrq!) It could even simply redirect to Inverse (mathematics).

This split would mirror, for example, the split between Reflection (mathematics) and Euclidean plane isometry. Melchoir (talk) 06:46, 23 May 2008 (UTC)

I'll be away for a bit. If there's no opposition by June then I'll carry out the split. Melchoir (talk) 01:44, 24 May 2008 (UTC)
 * As a test I've included inversive ring geometry on Inverse (mathematics) to check acceptability. There are 9 languages linking this Geometry article now considered for Move; some caution is in order. My concern is the tendency to raise an article's level of discourse too high for the general reader, especially the tendency to higher dimensions. This elite writing discredits WP. It is visible for instance at Poincare disk model and Poincare half-plane model, where complaints are useless. Would the proposed split result in one very low level article and a second elite one? I enjoy the mixed flavor of what we have now. Instead of going for higher dimensions for sophistication, one can look at the planar inversion in a hyperbola. Thus the two-dimensional situation need not mean the usual "circle" when one considers Unit sphere. It is very long articles that need splitting; here we have a modest beginning.Rgdboer (talk) 19:41, 24 May 2008 (UTC)
 * I think that very little of this article would be exported in a split. Possibly one could think of my proposal as a simple Move plus the creation of a stub. What makes it more of a split is the fact that some redirects would be sent to different destinations. For example, Inversive geometry should redirect to the broader article.
 * I don't know anything about inversion in a hyperbola -- what does one do with the asymptotes, or are they irrelevant? Anyway, there is plenty of room for sophisticated generalizations (in clearly-marked sections) in both articles. I suspect we agree that an article should introduce the base example first and point to generalizations later: this way one gets the "mixed flavor" without being too jumbled.
 * If the package proposal is too aggressive, shall we start by agreeing that "Circle inversion" is a better name for the article we have? Melchoir (talk) 23:46, 24 May 2008 (UTC)
 * The best name is Inversive Geometry since spheres are already used at two points. I understand that transformation geometry can run away from the elementary instance; the circle inversion is central to all expanded views, and stands well in the first section.Rgdboer (talk) 02:56, 27 May 2008 (UTC)
 * Yes, I suppose that works. Melchoir (talk) 19:51, 1 June 2008 (UTC)
 * Okay. Have made the move and updated links except Tetrahedral symmetry (couldn't find link) and perhaps some in Problem of Apollonius. Spotted work to be done.Will remove split tag now.Rgdboer (talk) 21:19, 2 June 2008 (UTC)
 * Thanks! I suggest though, that while Circle inversion redirects here, articles that wish to link the phrase "circle inversion" should link to the redirect, per WP:R2D. Melchoir (talk) 00:55, 3 June 2008 (UTC)

Proposed removal of a link
Klaus Th Ruthenberg has posted a link to his own website. The website is not relevant at all, I propose to remove it. Jbogaarts (talk) 11:27, 2 August 2009 (UTC)
 * Did you mean the "Natural geometry" ? It came up dead for me so now its gone. Go ahead and delete irrelevant links.Rgdboer (talk) 21:07, 2 August 2009 (UTC)