§ 10: The fundamental theorem

FS: Let A, B, C be three distinct points on a line l, and A', B', C' three distinct points on a line m (possibly l = m). Then there exists exactly one projectivity from l onto m, say φ, that maps A to A', B to B', and C to C'.

Proof: We already proved φ exists in O34.
Now suppose φ1 and φ2 meet our requirements, and let X be an arbitrary point of l.
Then we have (A', B'; C', φ1(X)) = (φ1(A), φ1(B); φ1(C), φ1(X)) = (A, B; C, X) = (φ2(A), φ2(B); φ2(C), φ2(X)) = (A', B'; C', φ2(X)).
According to O38 we find φ1(X) = φ2(X).
Since X was arbitrary it follows that φ1 = φ2.

Proposition: A bijection ψ : lm is a projectivity if and only if ψ preserves cross ratio.

Proof: We saw in the last section but this one that any projectivity preserves cross ratio.
Inversely, let ψ be a bijection from l onto m that preserves cross ratio.
Choose three points A, B, C on l, and let φ be the projectivity from l onto m with φ(A) = ψ(A), φ(B) = ψ(B), φ(C) = ψ(C).
Then we find for all X on l: (φ(A), φ(B); φ(C), φ(X)) = (A, B; C, X) = (ψ(A), ψ(B); ψ(C), ψ(X)) = (φ(A), φ(B); φ(C), ψ(X)).
According to O38 we have φ(X) = ψ(X). So φ = ψ. So ψ(X) is a projectivity.

Proposition: Each projectivity φ : lm is induced by a regular linear transformation of ℜ3.

Proof: P2 is the plane {x3=1} in ℜ3, extended with the points at infinity on the line at infinity.
Take three points A, B, C on l, and let A', B', C' be the image points on m under the projectivity φ.
Denote the vector from O to B by b, etc.
Then c = λa + μb en c' = ρa' + σb' for certain real numbers λ, μ, ρ, σ ≠ 0.
Let F be a regular linear transformation of ℜ3 with F(a) = (ρ/λ)a', F(b) = (σ/μ)b'.
Then F(c) = Fa + μb) = λF(a) + μF(b) = ρa' + σb' = c'.
In O41 we prove that F preserves cross ratio; or we'd better say that the bijection f from l onto m, induced by F, does so. Hence, according to the previous proposition, f is a projectivity from l onto m.
Since f(A) = φ(A), f(B) = φ(B), f(C) = φ(C), we get f = φ because of the fundamental theorem.


O39 Formulate and prove the dual of the fundamental theorem. (Hint: consider a straight line p, not through L or M, and the intersection of p and the pencil of lines L and M respectively.)

O40 Let φ be a projectivity from l onto m, and suppose the intersection point of l and m is invariant. Prove that φ is a perspectivity. Formulate the dual proposition, too.

O41 Let F be a regular linear transformation of ℜ3. Suppose that the end points of the vectors a, b, c are collinear. Prove that the end points of the vectors F(a), F(b), F(c) are collinear as well, and that we have:
|| (F(c) - F(a))|| / ||(F(b) - F(a)) || = || (c - a) || / || (b - a) ||. Subsequently, prove that the mapping f in P2 induced by F preserves cross ratio. (Project from O.)

O42 Using projective coordinates, let l: λ(1,0,0) and m: λ(0,1,0) be lines, and let A: λ(0,1,1), B: λ(0,2,1), C: λ(0,3,1) be points on l, and let A': λ(-1,0,1), B': λ(1,0,1), C': λ(0,0,1) be points on m.
Let F be the linear mapping from the proof of the third proposition in this section.
Check that λ=-1, μ=2, ρ=1/2, σ=1/2 and that (A, B; C, X) = (f(A), f(B); f(C), f(X)).