PROJECTIVE GEOMETRY COURSE

*Chapter 2*: PROJECTIVE MAPPINGS FROM A LINE ONTO A LINE

§ 8: *Perspectivities and projectivities*

Let *l* and *m* be two lines in the real projective plane, and let *P* be a point, not on *l* and not on *m*.

By projection from *P* we get a bijection from *l* onto *m* whose intersection point *l*.*m* is a fixed point.

We call this bijection a *perspectivity*, and use the formula *l*(*A _{1}*,

We call

We have at the same time a bijection from (the pencil of lines through)

We call the inverse mapping an

So the perspectivity from

Now the dual case: start from two points *L* and *M* and a line *p* not going through *L* or *M*.

The product of the elementary perspectivity of intersection from *L* to *p* and the elementary perspectivity of connection from *p* to *M* is a bijection from *L* to
*M*, which we call *perspectivity* with axis *p*.

The line *LM* is invariant. We write *L*(*x*,*t*,...) _{∧}=^{p} *M*(*x'*,*t'*,...), etc.

We call the composition of one or more (elementary) perspectivities a *projectivity*.

So we distinguish projectivities between ( the pencils of points on) two lines, projectivities between (the pencils of lines through) two points, projectivities between a line and a point,
and projectivities between a point and a line. An example:

Notations *l*(*A _{1}*,

This last formula means: "There exists a projectivity from

As an exercise, take an arbitrary point

Let

The mapping

Nota bene: formulas like *l*(*A*,*B*,*C*) _{∧}= *m*(*A'*,*B'*,*C'*) denote assertions (in this case: "there exists a perspectivity from
*l* onto *m* that maps *A* to *A'*, *B* to *B'*, and *C* to *C'* ").

When *A*, *B*, *C*, *D* lie on *l*, and *E*, *F*, *G*, *H* on *m*, the assertions *l*(*A*) _{∧}= *m*(*E*) and
*l*(*A*,*B*) _{∧}= *m*(*E*,*F*) are always true, but *l*(*A*,*B*,*C*) _{∧}= *m*(*E*,*F*,*G*) only if
*CG* goes through the intersection point of *AE* and *BF*;

the assertions *l*(*A*) _{∧}- *m*(*E*) , *l*(*A*,*B*) _{∧}- *m*(*E*,*F*) , and even
*l*(*A*,*B*,*C*) _{∧}- *m*(*E*,*F*,*G*) (see *O34*) are always true, but
*l*(*A*,*B*,*C*,*D*) _{∧}- *m*(*E*,*F*,*G*,*H*) is only true if *H* has a special position with respect to the other seven points.

*O32* Let *r* be a line, and let *R*_{∞} be the point at infinity of that line. Let *A*, *B* and *I* be distinct points on *r*, not equal to
*R*_{∞}, and *P* a proper point not on *r*.

Project the given points from *P* onto a line *r'* (going through *I* and not coinciding with *r*, with point at infinity __R___{∞}).

Also project *R*_{∞} onto *r'* and __R___{∞} onto *r*.

*O33* Choose three distinct points *A*, *B* and *C* on a line *l*. Let *P* be a point outside *l*, and *m* a line through *A* but not through *P*.

Project *A*, *B*, *C* from *P* onto *m* and find image points *A'*, *B'*, *C'*. Let *Q* be the intersection point *B'C.BC'*.

Show that there exists a projectivity satisfying *l*(*A*,*B*,*C*) _{∧}- *l*(*A*,*C*,*B*).

Make the dual construction, too.

*O34* Let *A*, *B*, *C* be three distinct points on a line *l* and *A'*, *B'*, *C'* three distinct points on a line *m*.

Prove that there exists at least one projectivity that maps *A*, *B* and *C* to (respectively) *A'*, *B'* and *C'*.

Distinguish three cases.