Talk:Cross-ratio

A tad technical what?
Isn't this all a bit technical here?

Of course one has to really really admire its contributing editors' grasp of elementary freshman math but the fact of the matter is that the curious 'intelligent lawyer' will come away from the page absolutely none the wiser.

Rather a pity don't you think for such an attractive topic in mathematics? I wonder what the late, great (and very unassuming) H.S.M. Coxeter would have thought of it all?

Of course I can appreciate that topics like the Riemann zeta-function don't easily lend themselves to popularisation but that's (frankly) not what we're talking here and in any case their editors do try and succeed in remaining encyclopaedic as well. Would these editors had taken time off to do the same here. I've added a 'technical' template.

I'm not aware that Pappus wrote on the cross ratio and have added 'citation needed' templates. At the very least it's an uncommon assertion which should be sourced. Rinpoche (talk) 21:02, 13 September 2010 (UTC)


 * Thank you for your input. As one of the contributing editors I thank you humbly for complimenting me on my excellent elementary freshman math skills. I try to stay in shape, you know, doing my morning integrals and all that. Would you care to explain why would an intelligent lawyer have any interest in cross-ratio? This is not something that comes up outside of mathematics at all. I doubt that scientists even in such quantitative fields as physics and chemistry get to see it. It is, however, an important topic in pure mathematics, and as such, it requires some background knowledge to appreciate, just as lawyers need their legal theory and doctors their anatomy. Arcfrk (talk) 04:38, 14 September 2010 (UTC)

Hello to the lawyer I congratulate you on your interest in this topic. The geometrical-cross ratio is of great technical and educational interest, and it can be considered in a quite elementary way, (its derivation is proved by similar triangles), or from the point of view of more generalized concepts. I believe that in any subject, however advanced the possible developments, it is important to always keep the educational path in sight, and I can see no good reason to try to obscure that path. I made a short contribution, a bit farther down in the discussions, suggesting how cross-ratios might be introduced. — Preceding unsigned comment added by 109.155.46.75 (talk • contribs) 00:46, 24 November 2014 (UTC)

4-tuple on a conic
The cross-ratio is defined for a 4-tuple of points on a conic in the real projective plane, by replacing such a 4-tuple by the 4-tuple of lines emanating from a fixed point on the conic, and passing through the 4 points. Does anyone know in what generality this can be done? Does this still work in the complex projective plane? Is there a source? Tkuvho (talk) 00:38, 17 December 2010 (UTC)
 * George Salmon (1900) A Treatise on Conic Sections, page 252. Real case.Rgdboer (talk) 03:54, 13 January 2011 (UTC)

The definition section
I just removed the following as it's not part of any definition of the cross ratio I've seen and put at the start of the section was confusing: it made it look like it was the definition, not the actual definition that followed, which does not depend on the division ratio. The extra detail about projective harmonic conjugates was also unnecessary: it's a special case for the cross ratio and so not part of the general definition.-- JohnBlackburne wordsdeeds 15:17, 13 February 2011 (UTC)


 * Given points a and b on an affine line, the division ratio of a point x is
 * $$d(x) = \frac {x - a} {x - b} .$$
 * Note that when a < x < b, then d(x) is negative, and is positive outside of the interval. When $$d(x) + d(y) = 0$$, then x and y are projective harmonic conjugates with respect to a and b. Note that this condition is equivalent to $$\frac {d(y)} {d(x)} = - 1$$. The cross-ratio $$(y, x; a, b) = \frac {d(y)} {d(x)}$$ is minus one in this case.

You say "not part of any definition of cross-ratio I've seen". So you did not read the reference given. Statement of the division ratio explains why cross-ratio is also called double ratio. Furthermore, use of division ratio clarifies how projective harmonic conjugates are related. This special case of cross-ratio has traditionally been exploited by authors to bolster understanding of the four-variable function.
 * Rgdboer (talk) 19:18, 13 February 2011 (UTC)
 * I've not read that book, no, and there's no ISBN or link for it. The two books I have are by Lawrence Edwards and Semple and Kneebone. Both define it first geometrically as the ratio of lengths. Mathworld gives the algebraic definition, with three other references, and the other references or links here that I can check give one or the other. But perhaps more importantly it's perfectly clear as it is: the definitions,
 * $$(z_1,z_2;z_3,z_4) = \frac{(z_1-z_3)(z_2-z_4)}{(z_2-z_3)(z_1-z_4)}.$$ and $$(A,B;C,D) = \frac {AC\cdot BD}{BC\cdot AD}.$$,
 * do not depend at all on the above material; as simple divisions they hardly need breaking into two fractions to calculate. The special case when the cross ratio is -1 worth mentioning but as a special case, not part of the definition, and it too does not depend on the division ratio.-- JohnBlackburne wordsdeeds 20:13, 13 February 2011 (UTC)

Cross-Ratio Definition
I suggest that (A,B;C,D) is defined as (AC/CB)/(AD/DB), and not as the algeraic simplification of that expression. In words, it is defined as the ratio of (the internal ratio of AC and CB) and (the external ratio of AD and DB). Algebraic simplification of the resulting experssion hides the intention and the path. — Preceding unsigned comment added by 84.92.190.75 (talk) 00:58, 24 November 2011 (UTC)

I suggest that the cross-ratio (A,C;B,D) of points A,B,C,D in that order on a line, is defined as (AB/BC)/(AD/DC), and not as the algeraic simplification of that expression. In words, it is the ratio of (the 'internal' ratio of distances to B from A and from C, where B is within AC) and (the 'external' ratio of distances to D from A and from C, where D is outside AC). Algebraic simplification of the resulting expression hides the intention and the path. Learning this example first, students can later learn that the four points have 6 possible cross-ratios, all algebraically related. — Preceding unsigned comment added by 86.176.40.219 (talk) 23:48, 19 September 2012 (UTC)

‎:We go by what reliable sources say on the subject. Are there references that use your proposed definition? Deltahedron (talk) 06:32, 20 September 2012 (UTC)

My point is simply about presentation and explanation, not an alternative formulation or derivation. One reference for the subject is "Geometry For Advanced Pupils" by E A Maxwell, 1920. — Preceding unsigned comment added by 86.182.171.234 (talk) 00:51, 29 September 2012 (UTC)
 * And what definition does that reference use? Deltahedron (talk) 06:37, 29 September 2012 (UTC)

I'm sorry - I should not have used the word "definition". My note is about a suggested "introductory explanation", to get a first grip on the subject, ignoring the issue of directed lengths, before a rigorous and general definition. E A Maxwell, in "Geometry For Advanced Pupils" (Oxford University Press, 1949), does show the cross-ratio algebraically as a ratio of ratios. I can not reproduce his printed layout but he introduces the notation (A,B;C,D) as AC over CB (horizontal division line), then a two line deep oblique division line, then AD over DB. That is, as opposed to the simplified form AC.DB over CB.AD. Before that he introduces a convention of directed lengths between the points. The suggestion of starting with an example where internal and external ratios are evident is mine, taken from other school geometry, but Maxwell does not start with such an example. Maxwell does not highlight a short definition section, but introduces (A,B;C,D) as above in his 5 pages of introductory text, in which he goes on to emphasises the importance of the order of letters, and then immediately shows the 6 possible values which permutations produce, namely m,  1-m,  1/m,  1/(1-m),  (m-1)/m, and  m/(m-1). His teaching strategy seems good.


 * The strategy can also be seen in Dirk Struik (1953) Lectures on Analytic and Projective Geometry, page 7, a reference once used on this page but removed since by an editor. See it used currently in projective harmonic conjugate where one also sees that midpoints of segments benefit from the double ratio viewpoint. In more recent geometry, particularly the shape of a triangle is expressed as a simple ratio, such as in the work of Rafael Artzy cited there. On the other hand, the Lester reference uses degenerate crossratio (involving ∝ ) to define the  shape of triangle. In the Lester-Artzy approach, cross-ratio is a ratio of the shapes of two triangles sharing a side.Rgdboer (talk) 02:09, 1 October 2012 (UTC)

A symmetric function on the cross ratio
I seem to remember there being a nice rational functional that you could plug the cross ratio into and it would return the same value for each of the 6 possible cross-ratios. Was that section removed or was I dreaming? — Preceding unsigned comment added by 129.94.176.102 (talk) 07:32, 15 August 2012 (UTC)
 * See Modular lambda function for the formula
 * $$ \frac{(1-\lambda+\lambda^2)^3}{\lambda^2 (1-\lambda)^2} \ . $$
 * Deltahedron (talk) 10:23, 18 August 2012 (UTC)

Cross-ratio and fundamental theorem of Galois, example 3
The elements of the group described in example 3 of the Fundamental theorem of Galois take the same form as the "six cross-ratios as Möbius transformations." I think they should be cross referenced somehow. — Anita5192 (talk) 22:05, 8 May 2015 (UTC)

Clifford Algebras
It should be mentioned a an analoge result in more dimensions of the cross ratio involving Clifford Algebras (these are as well a generalization of quaternions), that is useful for example to know when four points in $$R^d$$ belong to the same circle. (See for example: ) — Preceding unsigned comment added by 157.253.136.201 (talk) 22:10, 23 May 2016 (UTC)

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 * Checked and removed as not an especially valuable link worth keeping.-- JohnBlackburne wordsdeeds 20:05, 2 December 2016 (UTC)

Inconsistent italics?
It looks like symmetric groups are sometimes written like $$\mathrm{S}_3$$ and sometimes like $$S_3$$. Which is correct, assuming these are indeed intending to represent the same thing? -- Beland (talk) 19:01, 3 February 2023 (UTC)


 * It shouldn't really matter as long as it's not internally inconsistent in an article. The italic one is probably more common. YMMV. –jacobolus (t) 00:14, 4 February 2023 (UTC)

cross-ratio
To: Daniel Lazard: changes I have made (user 0ctavte0) adding an alternative definition of the cross-ratio and comments about more general approaches have been reverted by you, (from reading the history). I do hope you can write a short comment to make me understand if the reason was connected to mathematics or to procedures. Article has an obsolete approach and does not point to the core of the issue: Plucker constraints and billinear coordinated. 0ctavte0 (talk) 17:37, 13 May 2024 (UTC)


 * See my answer on your talk page. (the revert is motivated by Wikipedia policy WP:OR.) D.Lazard (talk) 17:51, 13 May 2024 (UTC)
 * OK. I understand. Policy is not debatable. It's law.
 * Can you recommend a person that might be interested in the subject and actually read the math? Maybe give advise how to proceed to publish. Someone that might want to get involved and be interested in co-authoring?
 * I am currently working on transferring the old results with second degree projective cuts (conics) to new structures. This is partly original work, not found in usual textbooks. But the ideas are pretty simple and put together old results in modern geometric objects, connect old coordinate computations with general theorems.
 * Along these ideas, to my opinion, the whole article on cross-ratio needs a root up review. 0ctavte0 (talk) 22:40, 13 May 2024 (UTC)
 * I am very sorry about using e-mail to message, I realized I was making this mistake just seconds after I have pushed the send button. Again, so sorry, indeed! And, as a proof, I have instantly switched to the talk page, as you have already seen.   Mea Culpa, Mea Culpa, Mea Maxima Culpa! 0ctavte0 (talk) 21:56, 15 May 2024 (UTC)
 * @0ctavte0 Any section like this needs to be supported by "reliable sources"; a self-published PDF doesn't cut it. Can you point to a published monograph, peer-reviewed journal paper, or similar? –jacobolus (t) 18:31, 13 May 2024 (UTC)
 * I understand.
 * On the other hand, I would highly and greatly appreciate some help along the lines in the reply above to professor Daniel Lazard. 0ctavte0 (talk) 22:43, 13 May 2024 (UTC)