How would I prove that the distance formula for points in the projective plane is invariant under an affine transformation?
Proof of Invariance for Distance Formula under Affine Transformation on Projective Plane
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linear-algebra
proof-writing
computational-geometry
invariance
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1What does it mean for a formula to be invariant if the quantity it computes is not invariant? Can you elaborate? – 2017-01-16
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0Or perhaps there is a converse distinction, that the formula might not be invariant though the quantity it computes is (under e.g. an affine transformation)? I don't know a way to distinguish a different notion. – 2017-01-17
1 Answers
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Distance is not invariant under affine transformations. Affine transformations preserve only the simple ratio, i.e. $(ABC):=\frac{AC}{CB}$ (the quotient of signed distances), where $A$, $B$, $C$ are collinear.