WT37: The cross ratio

[benthurston27]

Can the cross ratio always be written as a positive number? If so I don't see why your proof of the invariance of R(A,B,C,D)^2 is any weaker because you don't lose solutions.

[njwildberger]

@benthurston27 No: the cross ratio can have either positive of negative values.

[benthurston27]

@njwildberger I think you pointed out in the next video if a certain labeling of the points produces K, a different labeling will produce 1-K. So if K is -1 then 1-K=2. I guess I still don't understand what logical difference it makes the choice of the labeling?

[njwildberger]

@benthurston27 It is an important and interesting point, the ambiguity in the cross ratio depending on the order of the four points we take.

[MrFunatabi]

I mention in passing that if you felt uncomfortable with using the triangle proportions theorem in this situation, as I did, you can derive the R(A, B : C, D)^2 formula with the spread law. (ratio of a spread and the opposite quadrance is the same in a triangle)

This is because the law of proportion theorem itself is derived from the spread law. See WT9 for the proof of "the laws of proportion for a triangle."