COURSE OF PROJECTIVE GEOMETRY

§ 9: answers

*O35*

i) P at infinity, D proper.

Draw a help line through *A* parallel to *l'*.

ii) *P*, *D*, *D'* at infinity.

As i), but now we have only to prove *AC/BC* = *A'C'/B'C'*.

iii) *D* at infinity, *P* proper.

This is the limit of the general case.

iv) *l* line at infinity.

In this case, the proposition follows from the definition of the cross ratio of four points on the line at infinity.

*O36*

1 2 3 4 λ 2 1 3 4 μ 3 1 2 4 ν 4 1 2 3 ρ

1 2 4 3 μ 2 1 4 3 λ 3 1 4 2 σ 4 1 3 2 τ

1 3 2 4 σ 2 3 1 4 τ 3 2 1 4 ρ 4 2 1 3 ν

1 3 4 2 ν 2 3 4 1 ρ 3 2 4 1 τ 4 2 3 1 σ

1 4 2 3 τ 2 4 1 3 σ 3 4 1 2 λ 4 3 1 2 μ

1 4 3 2 ρ 2 4 3 1 ν 3 4 2 1 μ 4 3 2 1 λ

where μ=1/λ, ν=1/(1-λ), ρ=λ/(λ-1), σ=1-λ, τ=1-1/λ.

*O37* If *X*_{1}, *X*_{2}, *X*_{3}, *X*_{4} are four points on a line *l*, and
*x*_{1}, *x*_{2}, *x*_{3}, *x*_{4} four lines such that *x*_{i} has the same projective coordinates as
*X*_{i}, then *x*_{1}, *x*_{2}, *x*_{3}, *x*_{4} go through one point *L* with
the same projective coordinates as *l*:

*x*_{i} is the line of intersection of x_{3}=1 and the plane α_{i} through *O* whose normal is *OX _{i}*; these four planes go through the normal of
the plane through

Let

*O38* Consider the line as an axis of the real numbers, so f(x) = E(x-c)/(x-d) with E=(b-d)/(b-c). Draw the graph of f.

For each point of *l* (proper or at infinity), there exist a vicinity *V* of the image of that point and a vicinity *U* of the point itself such that if Y ∈ *U* then
f(Y) ∈ *V*.