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:
- .
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
- .
Q.E.D.
Read more about this topic: Duality (projective Geometry)
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