Talk:Inversive geometry

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)

a small mistake about notation.

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)

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 (talk • contribs)

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)

Bad diagram

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)

Removal of pole & polar

I have temporarily hidden the section that was recently added concerning poles and polars. What was described in this section is called reciprocation in a circle, a Euclidean concept. It is only a partial function in the Euclidean plane as it is not defined for the center of the circle or for any of the lines which pass through the center. It does not extend to a function in the inversive plane (it does extend to a duality in the real projective plane). Due to this, it is not a topic that is generally brought up in inversive geometry. Its inclusion in a prominent position in this article is a mistake. At best, reciprocation can be brought up as an application of inversion in the Euclidean plane, but you will be hard pressed to find a citation that mentions poles and polars in the inversive geometry context. I have hidden it rather than removing it thinking that someone might want to try to find an appropriate way to incorporate this material into the article, but my preference would be to just remove it. Bill Cherowitzo (talk) 15:26, 21 May 2015 (UTC)

I added that section , inversion points and polars hang close together so I thought it would be a good extra. for example in Inversion in circle.svg ,the second picture N' and N are on the polar of pole P. Circle inversion itself is also not an involution, it needs an extra point at infinity, (the pole of a diameter is like wise a point at infinity only with a polar it has a direction direction orthogonal to the diameter)

Maybe you are right that it would be better under circle inversion # application , but then it does deserves a seperate section, and I don't know under which subsections I could put the other parts of circle inversion # application. WillemienH (talk) 16:34, 21 May 2015 (UTC)

Why?

Could/should there be a section, "Uses of inversion" or "Usefulness of inversion" or "Applications of inversion"? For non-mathematicians, it would be great to have a simple explanation of why and how inversion is useful, and/or why it matters. It does say "Many difficult problems in geometry become much more tractable when an inversion is applied." but that's pretty vague. There is an "Application" section under Circles but it's clearly written for people who already know a lot of higher-order math, and doesn't really explain anything to the lay person. Bookgrrl ^holler/lookee here 17:12, 12 March 2017 (UTC)

+1. 67.198.37.16 (talk) 04:23, 8 May 2019 (UTC)