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There are I think at least two views on the cross ratio of 4 points even https://en.wikipedia.org/wiki/Cross-ratio gives two definitions

And I was wondering what is the relation between the two

The Geometrical view:

The cross ratio of 4 points A,B;X,Y is

$$ CR_g =\frac{d(A,Y) d(B,X)}{d(A,X) d(B,Y)}$$

Where d(I,J) is the distance between I and J.

The Algebraic view:

The cross ratio of four (complex) numbers a,b,x,y is:

$$ CR_a=\frac{(y-a) (x-b)}{(x-a)(y-b)}$$

Wondering / Questions

Points in a plane can be represented by complex numbers and this made me wonder:

What is the relationship between the two cross ratio views if $A,B,X,Y$ and $a,b,x,y $ represent the same points.

And I also would like to see a proof of that relation.

Explorations:

When the points A,B,X,Y are on a line then the cross ratio's are equal

When the points are on a circle $CR_a$ is a real number (see reference in comments)

Willemien
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  • Depending on your familiarity with the complex plane and Mobius transformations, this answer probably sums it up most succinctly (+1 for the question and exploration thereof, btw). – dxiv Jun 02 '17 at 05:02
  • Thanks by the way the proof is only that $CR_a$ is real it doesn't proof that $CR_a\text{4 points on circle} = CR_g {same points} $ – Willemien Jun 02 '17 at 06:21
  • But yes, it does, since $d(X,Y)=|x-y|,$, so if $CR_a$ is real then $|CR_a| =CR_g,$. – dxiv Jun 02 '17 at 06:26
  • I am exploring $d(X,Y)=|x-y| $ at the moment seems to open another can of worms found two different (unequal?) equations for $|x-y|^2$ – Willemien Jun 02 '17 at 06:34
  • @Willemien I don't even see how $CR_g$ is a valid representation of the cross ratio. You can't even get negative values out of it, so there will never be harmonic sets. It would make sense to use a signed distance, and then I don't think there is any difference. – rschwieb Jun 06 '17 at 13:31
  • @rschwieb both definitions are in https://en.wikipedia.org/wiki/Cross-ratio and also for $CR_a$ you need a coordinate system while for $CR_g $ you only need a distance function – Willemien Jun 07 '17 at 01:39
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    The Wikipedia article says 'cross ratio (...) is a number associated with a list of four collinear points' and, additionally, 'where an orientation of the line determines the sign of each distance'. – CiaPan Jun 07 '17 at 07:45
  • @Willemien Well, if I'm remembering correctly, you need a Desarguesian projective plane to talk about cross ratios. Then in theory the Desarguesian plane determines a division ring that coordinatizes it (and any lines it contains.) So the two are equivalent (although there is a great deal of work to make the connections plain.) CiaPan's comment is a good followup to what I was trying to say. – rschwieb Jun 07 '17 at 13:34

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