WT37: The cross ratio

The cross ratio is the most important invariant in projective geometry, and plays a key role in hyperbolic geometry. We introduce it here using vectors, that is in the framework of affine geometry.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.

Category:

Science & Technology

Tags:

• Mathematics
• foundations
• geometry
• cross
• ratio
• rational
• trigonometry
• projective
• hyperbolic
• vectors
• affine

[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."