# Talk:Inversive geometry

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Field: Geometry

## QBAsic

Isn't the QBasic program a bit out of place? Hardly anyone uses QBasic anyway. Phys 05:31, 1 Jan 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

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)

## 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.

## 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)

## 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.

## 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)

## 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)

## 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#Quadratic forms. 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)

## Orthogonality of spheres?

In the section Inversive geometry and hyperbolic geometry, does the expression "orthogonal to the unit sphere" mean that the two spheres are orthogonal at their tangent planes are orthogonal at the points of intersection? If so, what if the spheres don't intersect? LachlanA (talk) 06:13, 30 May 2008 (UTC)

First question: Yes. Second question: Nonintersecting spheres are not considered to be orthogonal and aren't considered lines in the hyperbolic space.--RDBury (talk) 05:18, 13 September 2009 (UTC)

## reciprocation and Möbius-Group

Please review the paragraph on "transformation theory/reciprocation". reciprocation is not an element of $\text{Aut}(\hat{\mathbb{C}})$. Rather $z\mapsto 1/z$ ("algebraic" inversion) is in (and a generator of) the Möbius-group, with Matrix-representation $\begin{pmatrix}0 &i \\i &0 \end{pmatrix} \in \text{SL}(2,\mathbb{C})$ and fixed points $\pm 1$. It is not inversion-in-a-circle, rather reciprocation is. Or correct me if i'm wrong. Nesta 4iver (talk) 07:15, 4 November 2008 (UTC)

The paragraph states that reciprocation is the composition of conjugation with inversion-in-unit-circle. Inversive geometry is richer than Mobius geometry since all three of these mappings fall in its reach. Usually Mobius geometry includes z --> 1/z but not the angle-reversing maps conjugation and circle-inversion.Rgdboer (talk) 21:12, 4 November 2008 (UTC)

In the section "Inversive geometry and hyperbolic geometry", the formulas lack foot note. $x_{1}^{2}+\cdots +x_{n}^{2}+2a_{1}x+\cdots +2a_{n}x+c=0$Liuyifourfire (talk) 08:04, 31 March 2009 (UTC)

Yes, sharp eyes Liuyifourfire. The subscripts have now been inserted to make the formula appear thusly:
$x_{1}^{2}+\cdots +x_{n}^{2}+2a_{1}x_1+\cdots +2a_{n}x_n+c=0$
The encyclopedia will be brought to perfection by such attention to detail.Rgdboer (talk) 21:30, 2 August 2009 (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)

## Terminology standard

The terms 'reference circle', 'inversion circle', and 'circle of inversion', all meaning the same thing, are used with pretty much equal frequency by various source. 'Circle of inversion' seems to be the original term and is more usual in older, more geometrical sources, while 'reference circle' seems to be used by the more trendy, computer graphics type sources. Any thoughts on what should be used here? Sticking with what's already there is a strategy too but there are other options.--RDBury (talk) 10:51, 13 September 2009 (UTC)

## The basic inversion equation is lacking

I am not qualified to make changes to this page, but for a layman trying to get a grip on the matter I feel that the most basic and important info of all is lacking--i.e. how you invert an equation/shape.

I would propose that this info be included up near the top somewhere--

The point $(x,y)$ inverted is: $\left(\frac{x}{\sqrt{x^2+y^2}}, \frac{y}{\sqrt{x^2+y^2}}\right)$

Perhaps it would also be nice to include a simple example--for example, the inversion of an ellipse into a limacon:

The ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ inverts to the limacon $\frac{(x+b(x^2+y^2))^2}{a^2}+\frac{y^2}{b^2}=(x^2+y^2)^2$ —Preceding unsigned comment added by 190.24.55.241 (talkcontribs)

Actually what you've written is incorrect. Here's the inversion in the unit circle centered at (0, 0):
$(x,y) \mapsto \left( \frac{x}{x^2 + y^2}, \frac{y}{x^2 + y^2} \right)$
But this analytic way of looking at it is not necessarily the whole story. Michael Hardy (talk) 21:02, 1 December 2009 (UTC)
PS: If x is a vector with two or more scalar components, then the above can be written as
$x \mapsto \frac{x}{|x|^2}.$
That's pretty close to some of the things that are in the article. Michael Hardy (talk) 21:05, 1 December 2009 (UTC)
I think much of what you're looking for is in the article Inverse curve. You could make an argument for merging the two, but both are fairly large already and they cover different aspects of the subject.--RDBury (talk) 08:30, 2 December 2009 (UTC)

I have removed the following diagram:

A procedure to construct the inverse P' of a point P outside a circle C. Let r be the radius of C. Since the triangles OPN and OP'N are similar, OP is to r as r is to OP'.

While the caption is correct, the diagram does not show the true location of P'. There is an extraneous circle drawn with center N ... this is not needed for anything. P' is located at the intersection of OP and the line connecting the points of tangency of the tangents drawn from P to the reference circle. When I get a chance I'll put a corrected diagram back in. Bill Cherowitzo (talk) 21:36, 30 November 2011 (UTC)

Diagrams like this should use SVG rather than PNG format as well.--RDBury (talk) 08:28, 1 December 2011 (UTC)

I agree, unfortunately I ran into a technical difficulty that I wasn't expecting ... Open Office would not convert the labels when it exported to SVG. I did upload the unlabelled version as "Inversion in cirle.svg", if anyone wishs to add the labels and replace the file, please do so. Also, I was a bit hasty in claiming that the diagram was incorrect ... it actually is fine. I did replace it with a simpler construction with a more obvious proof, and it can be made to work with P inside the circle as well. Bill Cherowitzo (talk) 05:11, 2 December 2011 (UTC)

### New pic, new problem

I have a problem with the new version of this pic (http://upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Inversion_in_circle_2.png/800px-Inversion_in_circle_2.png). No reason is given why the triangles OPN and OP'N are similar. It's obvious to you math wizards, and they LOOK similar to the eye, but I do not see it in any rigorous way (other than that they're both right triangles and both use the radius as a side).
This encyclopedia is aimed at ordinary people, not math wizards. Even though I am profoundly, abysmally stupid; tests indicate that nevertheless, I am far less stupid than the ordinary person (for whom this article should be written). I therefore suggest that one of you explain in the caption why the triangles are similar.
Yes, I could go look it up myself, but we stupid people are also characteristically lazy.
Dave Bowman - Discovery Won (talk) 11:24, 2 January 2013 (UTC)
The caption space is getting a bit crowded. All I could do was add a hint about the proof. Perhaps it makes more sense to pull this out and write it up as a subsection. In circle with diameter OP, angles NOP and NPO are measured by half of their inscribed arcs. Since this is a semicircle, these angles are complimentary. The third angle of triangle NOP' is also complimentary to angle NOP (due to being in a right triangle) and so congruent to angle NPO ... giving the similarity of the right triangles. Bill Cherowitzo (talk) 20:40, 2 January 2013 (UTC)
But perhaps you were looking for something a bit simpler, such as the fact that angle NOP appears in both triangles, making them similar because they are right triangles. Bill Cherowitzo (talk) 16:41, 3 January 2013 (UTC)

## Cayley

In the section "Higher geometry" an edit was make to exclude Cayley as an originator of early models of non-Euclidean geometry. Cayley is known for the introduction of projective metrics, a method that is now standard. His "Sixth Memoir" (1859) is celebrated. As John Wesley Young wrote (pages 177–8) in Projective Geometry (1930), "That both the hyperbolic geometry of Bolyai-Lobachevski and the elliptic geometry of Riemann may be defined as geometries associatated with subgroups of the projective group was first shown by Arthur Cayley (1859), although he did not formulate his ideas on the basis of group theoretic considerations." Since attention here is on the early models, reference to Poincare is out of place. Often the Klein model is called the Cayley-Klein model, so it is not true that no model is associated with him.Rgdboer (talk) 21:27, 8 December 2011 (UTC) See also Cayley-Klein metric for reference.Rgdboer (talk) 21:29, 8 December 2011 (UTC)

I removed Cayley since there is no model that is associated with him. It is his metric which was crucial for the development of Klein's model. Granted, Klein would not have come up with his model without Cayley's work but that does not (at least in my mind) legitimatize the attribution of Cayley-Klein to the model. They neither worked together, nor independently came to the same results – which is my criteria for combined names like this. Math history is filled with mislabeled results, so pointing to this name does not prove anything. If you want to make the claim that Cayley produced a model, find a reference that says he did ... not an ahistorical comment like Young's, nor an inference from a name. Klein's work was published in 1872, 13 years after Cayley's result and Poincaré's work follows Klein by only 10 years, so I do not see a great distinction between "early" and "not early" work. Bill Cherowitzo (talk) 01:10, 9 December 2011 (UTC)

## Conformal Symmetry

There should be some links between this topic and the pages on conformal symmetry, conformal group, conformal maps, conformal transformations etc. There is too much overlap and repetition without cross-referencing

It would also be good to explain links with twistor theory, and why the conformal group in d dimensions is spin(d,2) Weburbia (talk) 14:11, 7 July 2013 (UTC)

## Magnus or Steiner?

Near the bottom of page 77 in the Second Edition of Coxeter's Introduction to Geometry (Wiley Classics paperback) Coxeter states

"The transformation called inversion, which was invented by Jakob Steiner about 1828, is new ,,,"

not Magnus. Wolfram Mathworld also credits Steiner with priority. But I do not have a reference for Steiner's work. — Preceding unsigned comment added by 86.185.249.166 (talk) 12:12, 16 October 2013 (UTC)

According to Eves, A Survey of Geometry (Vol. 1), 1963 p. 145:

The history of the inversion transformation is complex and not clear-cut. [...] But inversion as a simplifying transformation for the study of figures is a product of more recent times, and was independently exploited by a number of writers. Bützberger has pointed out that Jacob Steiner disclosed, in an unpublished manuscript, a knowledge of the inversion transformation as early as 1824. It was refound in the following year by the Belgian astronomer and statistician Adolphe Quetelet. It was then found independently by L. I. Magnus, in a more general form, in 1831, by J. Bellavitis in 1836, then by ....

Eves does not provide references for any of this. My 2nd edition of Coxeter (hardback) has the Magnus reference on page 77, so he must have made the correction between printings of the book. I don't have a reference for the 1828 date. Bill Cherowitzo (talk) 16:24, 16 October 2013 (UTC)

The nineteenth century is rather late to be looking for the origin of this transformation. Refer to Pappus chain for a construction that pushes the notion back into antiquity.Rgdboer (talk) 20:06, 16 October 2013 (UTC)
Julian Coolidge (1940) in his book History of Geometrical Methods has these relevant comments:
Page 65: if the products of their lengths is constant, we have a famous transformation called inversion that was not thoroughly recognized until the nineteenth century and whose paternity has been ascribed to many geometers.
Page 279: this transformation was first mentioned by Pappus, who knew that it carried a line or circle into a line or circle. It has been discovered subsequently by several writers, the first being perhaps Steiner.
Before Coolidge, it was Michel Chasles that immersed himself in the arguments of Pappus to such an extent that the ancient text has had a modern influence.Rgdboer (talk) 21:03, 17 October 2013 (UTC)