| 1. | These three and the identity permutation leave the cross ratio unaltered.
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| 2. | M�bius transformations satisfy exactly the same estimates, since M�bius transformations preserve the cross ratio.
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| 3. | In fact, every such invariant is expressible as a function of the appropriate cross ratio.
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| 4. | Could you give a proof that the cross ratio is the image of z _ 4?
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| 5. | I don't understand how the cross ratio answers when 4 points in projective line are projectivly equivalent.
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| 6. | The cross ratio of four different points is real if and only if there is a line or a circle passing through them.
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| 7. | Every four collinear points with rational coordinates have a rational cross ratio, so the Perles configuration cannot be realized by rational points.
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| 8. | :Afraid not, the cross ratio you get from joining opposite corners and projecting lines will be the same as for a square.
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| 9. | Modern use of the cross ratio in projective geometry began with Lazare Carnot in 1803 with his book " G�om�trie de Position ".
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| 10. | How can the cross ratio, a scalar, be the image of a map on the projective line, a point defined by two coordinates?
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