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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)
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)
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)
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 words deeds 15:17, 13 February 2011 (UTC)
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.
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)
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.
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)
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)
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 belong to the same circle. (See for example: [2] ) — Preceding unsigned comment added by 157.253.136.201 ( talk) 22:10, 23 May 2016 (UTC)
References
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It looks like symmetric groups are sometimes written like and sometimes like . Which is correct, assuming these are indeed intending to represent the same thing? -- Beland ( talk) 19:01, 3 February 2023 (UTC)
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)
This article is rated C-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
This page has archives. Sections older than 365 days may be automatically archived by Lowercase sigmabot III when more than 10 sections are present. |
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)
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)
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)
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 words deeds 15:17, 13 February 2011 (UTC)
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.
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)
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.
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)
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)
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 belong to the same circle. (See for example: [2] ) — Preceding unsigned comment added by 157.253.136.201 ( talk) 22:10, 23 May 2016 (UTC)
References
Hello fellow Wikipedians,
I have just modified one external link on Cross-ratio. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
When you have finished reviewing my changes, please set the checked parameter below to true or failed to let others know (documentation at {{
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).
An editor has reviewed this edit and fixed any errors that were found.
Cheers.— InternetArchiveBot ( Report bug) 19:56, 2 December 2016 (UTC)
It looks like symmetric groups are sometimes written like and sometimes like . Which is correct, assuming these are indeed intending to represent the same thing? -- Beland ( talk) 19:01, 3 February 2023 (UTC)
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)