COURSE OF PROJECTIVE GOMETRY

§ 1: answers

*O1*

iii) (see the picture above) *l'* and *m'* intersect in the point of projection of *P* on α (direction of projection *l*).

iv) (see the picture above) *l'* and *m'* are parallel to the line that connects *P* and the point of intersection of *l* and *m* on *s*.

*O2* All concepts mentioned above except distance and angular measure.

Example: if *l* and *m* in β are parallel, then so are their images *l'* and
*m'* under parallel projection onto α.

*O3* *P*→*P"* where *P"* is the other point of intersection of *T**P'* and *c* (*T* at the top of *c*).

The point *U* where *l'* intersects *u* has no original under the central projection, so the other point of intersection of *T**U* and *c* has no original.

Furthermore, *T* itself has no original, either.

After reading the next section you will understand that the central projection maps the point at infinity of *l* onto *U*, and that the mapping of this problem maps *S* onto *T*.

You can see that the mapping is continuous if you formulate continuity in terms of neighbourhoods (of the proper points and of the point at infinity on a line).

*O4* The tangents that touch *c* in *A* and *B* correspond to the asymptotes of the hyperbola.

The lines through *A* or *B* in β correspond to the lines in α that have an asymptotical direction.

Because these lines intersect *c* in at most one point that doesn't lie on *s*, their image lines intersect the hyperbola in at most one (proper) point.