PROJECTIVE GEOMETRY COURSE

*Chapter 1*: INTRODUCTION

§ 1: *Central projection*

Consider two non-parallel planes α and β and a point *P* that is not in any of both planes.

Let *s* be intersecting line of β and the plane through *P* parallel to α.

Let *t* be the intersecting line of α and β.

Let *u* be the intersecting line of α and the plane through *P* parallel to β.

For each point *X* in β that is not lying in *s*, let *X'* be the point of intersection of α and the line connecting *P* and *X*.

The mapping *X*→*X'* is a bijection from β/*s* onto α/*u*, named *central projection*. We find:

i) If the points *A*, *B*, *C* in β/*s* are collinear (on *l*), then so are the image points *A'*, *B'*,
*C'* (on *l'*).

ii) If the lines *a*, *b* and *c* in β concur in a point *L* of β/*s*, then the image lines *a'*,
*b'* and *c'* concur in *L'*.

iii) If *l* and *m* are parallel lines in β that intersect *s* in distinct points, then *l'*
and *m'* intersect in a point of *u*.

iv) If *l* and *m* are lines in β, parallel to *s* but not identical to *s*, or intersecting in a point of *s*,
then *l'* and *m'* are parallel.

v) Let d(*A*,*B*) be the distance between *A* and *B*. Suppose that *A*, *B* and *C* are collinear in β/*s*.
In general we have d(*A*,*B*) ≠ d(*A'*,*B'*) and d(*A*,*B*)/d(*B*,*C*) ≠ d(*A'*,*B'*)/d(*B'*,*C'*).

vi) Let ∠(*l*,*m*) be the smallest angle between *l* and *m*. Let *A*, *B* and *C* be non-collinear points in β/*s*.
In general, we find ∠(*A**B*,*B**C*) ≠ ∠(*A'**B'*,*B'**C'*).

vii) The image of a circle *c* in β is a conic section in α: an ellipse if *c* and *s* don't intersect, a parabola if *c*
touches *s*, and an hyperbola if *c* and *s* intersect in two points.

We provisionally define *projective geometry* as the study of those geometrical concepts that are invariant under central projection.

We have, among others, the following projective concepts:

a) points, lines, incidence of points and lines (a point *A* is incident with a line *l* if *A* lies on *l*);
collinearity and concurrence, complete triangles and complete quadrangles;

b) separation and cross ratio

(let *A*, *B*, *C*, *D* be four points on a line *l*, then the pairs {*A*,*B*} and
{*C*,*D*} separate if one of the points *C* and *D* lies between *A* and *B*, and the other doesn't;

the cross ratio
(*A*,*B*;*C*,*D*) is the number (d(*A*,*C*)/d(*A*,*D*)):(d(*B*,*C*)/d(*B*,*D*)), provided with a
minus sign if {*A*,*B*} and {*C*,*D*} separate), see section 9.

Concepts like distance and ratio of distances, measure of angles, ellipse, parabola and hyperbola, parallel, are *not* projective concepts (but the
concepts conic and touching are).

*Problems*:

*O1* Verify all assertions above.

*O2* When we use *parallel projection* instead of central projection, we replace the bundle of lines through *P* by a bundle of parallel lines that intersect α and β.

A geometrical concept that is invariant under parallel projection is called an *affine concept*. Which of the concepts mentioned above are zijn affine?

*O3* Let *l* be a line in β that intersects *s* in *S*, and *l'* its image line in α. Let *c* be a circle that touches *l'*.
Give the instructions for a continuous mapping from *l* onto *c*.

*O4* Suppose that the circle *c* in β intersects the line *s* in *A* and *B*, so that the image curve is a hyperbola.

Which lines in β are corresponding to the asymptotes of this hyperbola?

Explain by reference to the situation in β that the lines in α with asymptotical direction intersect the hyperbola in at most one point.