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The cross ratio relates the positions of four co-linear points in 3d space. I understand definitions a published online, such as that from wikipedia:

https://en.wikipedia.org/wiki/Cross-ratio

However, I'm am unsure what the general rule is for point selection in this ratio.

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  • $\begingroup$ The denominator is the (distances of the) inner pair times the outer pair, and the numerator is the product of the (distances of the) two overlapping pairs. Is that what you mean? $\endgroup$
    – almagest
    Apr 17, 2016 at 7:12
  • $\begingroup$ That is what I'm talking about, but I've seen many different forms of that relationship from different sources. $\endgroup$
    – Austin
    Apr 17, 2016 at 19:28
  • $\begingroup$ For example, given four co-linear points in order (A, B, C, D), then the cross ratio could be (AB * CD) / (AC * BD), or it could also be (AD * BC) / (AC * BD). $\endgroup$
    – Austin
    Apr 17, 2016 at 19:35

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It's mostly a matter of convention. In my surroundings, one would define

$$CR(A,B;C,D):=\frac{[A,C][B,D]}{[A,D][B,C]}$$

which is the same order Wikipedia uses. I assume that convention to be the most common one. But I, too, have seen other conventions, so it's best to check what convention a given source is using.

I'd use $[A,C]$ to denote a determinant of $\mathbb{RP}^1$ homogeneous coordinates, but others use other symbols here, sometimes $AC$ for the (oriented!) length, sometimes $A-C$ as a difference of (non-homogeneous) coordinates. Mostly a matter of notation and convention, once you've seen that all these things are essentially the same.

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