PROJECTIVE GEOMETRY COURSE

§ 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 φ_{1} and φ_{2} meet our requirements, and let *X* be an arbitrary point of *l*.

Then we have (*A'*, *B'*; *C'*, φ_{1}(*X*)) = (φ_{1}(*A*), φ_{1}(*B*); φ_{1}(*C*), φ_{1}(*X*)) =
(*A*, *B*; *C*, *X*) = (φ_{2}(*A*), φ_{2}(*B*); φ_{2}(*C*), φ_{2}(*X*)) =
(*A'*, *B'*; *C'*, φ_{2}(*X*)).

According to *O38* we find φ_{1}(*X*) = φ_{2}(*X*).

Since *X* was arbitrary it follows that φ_{1} = φ_{2}.

*Proposition:* A bijection ψ : *l* → *m* 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 : *l* → *m* is induced by a regular linear transformation of ℜ^{3}.

*Proof:* *P*^{2} is the plane {x_{3}=1} in ℜ^{3}, 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

Let

Then

In

Since

*Problems*

*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 ℜ^{3}. Suppose that the end points of the vectors * a*,

|| (

*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*)).