Duality (projective Geometry) - Preservation of Incidence

Preservation of Incidence

The duality mapping g is an isomorphism with respect to the incidence properties (such as collinearity and concurrency). The mapping g has this property: given a pair of lines L₁ and L₂ which intersect at a point P, then their dual points gL₁ and gL₂ define the unique line g−1P:

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Given points P₁ and P₂ through which passes line L, P₁.P₂ = L, then what is the intersection of lines g−1P₁ and g−1P₂? If g−1P₁ ∩ g−1P₂ = P then

so that

∴

Given a pair of affine points in homogeneous coordinates, the line passing through them is

where the cross product is computed just as it would for an ordinary pair vectors in 3-space.

From this last equation can be derived the intersection of lines, by using the mapping g to "plug in" the lines into the slots for points:

where mapping g is seen to distribute with respect to the cross product: i.e. g is an isomorphism of cross product.

Theorem. The duality mapping g is an isomorphism of cross product. I.e. g is distributive w.r.t. cross product.

Proof. Given points A=(a:b:c) and B=(d:e:f), their cross product is but

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Therefore

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Q.E.D.