:Cross-ratio is a projective invariant i . e . it is unchanged by projective transformations.
2.
The condition that " h " be non-degenerate seems to be a projective invariant.
3.
The cross-ratio is preserved by the fractional linear transformations and it is essentially the only projective invariant of a quadruple of collinear points, which underlies its importance for projective geometry.
4.
Collinearity is not the only geometric property of configurations of points that must be maintained for example, five points determine a conic, but six general points do not lie on a conic, so whether any 6-tuple of points lies on a conic is also a projective invariant.
5.
While the projective linear group of the plane is 3-transitive ( any three distinct points can be mapped to any other three points ), and indeed simply 3-transitive ( there is a " unique " projective map taking any triple to another triple ), with the cross ratio thus being the unique projective invariant of a set of four points, there are basic geometric invariants in higher dimension.
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