The Fundamental Theorem of Projective Geometry
If L =(A,B,C,D) and L' = (A',B',C',D') are lists of points of RP2, each with no three collinear,
then there is a unique element of P(2) mapping L to L'.
Much as in affine geometry, this follows a special case with the standard points
X = [1,0,0], Y = [0,1,0], Z = [0,0,1], U = [1,1,1] in one of the lists.
The (X,Y,Z,U) Theorem
If L =(P,Q,R,S) is a list of points of RP2, with no three collinear, then
there is a unique element of P(2) mapping (X,Y,Z,U) to L.
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Proof of the (X,Y,Z,U) Theorem
Let P, Q, R, S be the p-points [p], [q], [r], [s] respectively,
and let x, y, z, u
be the vectors (1,0,0), (0,1,0), (0,0,1), (1,1,1).
Observe that, as no three of P,Q,R,S are collinear, no three of
p,q,r,s are linearly dependent.
Suppose that t is defined by the matrix A.
Then t(U) = S if and only if [Au] = [s], i.e. Au = λs
with λ ≠ 0.
If we replace A by (1/λ)A, we get the same t, so we may assume that Au = s
Now, Ax, Ay and Az are just the columns of A (in order), so
t maps X, Y, Z to P, Q, R if and only if the columns of A are
of the form αp, βq, γr, with
α, β, γ ≠ 0.
Also Au is the sum of the columns of A so that t has the required images
if and only if αp + βq + γr = s.
Since p, q, r are linearly independent, this has a unique solution α, β, γ.
Since s is not dependent on any two of p, q, r,
α, β, γ are non-zero.
Thus the matrix A with columns αp, βq, γr is invertible, so defines
an element
of P(2). From the above argument, A is unique up to scaling, so t is unique.
Proof of the Fundamental Theorem
By the (X,Y,Z,U) Theorem, there exist elements r, s of P(2)
such that
r maps (X,Y,Z,U) to L and s maps (X,Y,Z,U) to L'.
Then t = sor-1 maps L to L'.
Suppose that u also maps L to L'. Then uor maps (X,Y,Z,U) to L'.
By the uniqueness clause of the Theorem, only s maps (X,Y,Z,U) to L'.
Thus uor = s, so u = sor-1 = t,
i.e. t is unique.
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