## Projective Geometry I - Lecture Notes

#### 6.1 The Fundamental Theorem: Axiom P7

**P7: (Fundamental Theorem for projectivities on a line)** Let *l* be a line in a projective
plane. Let A, B, C and A', B', C' be two triples of three distinct points on *l*. Then there is at
most one projectivity of *l* into *l* such that ABC __^__ A'B'C'.
**Rmk:** If P7 holds on one line, it holds on every line.

A projectivity not equal to the identity has at most 2 fixed points (since ABC __^__ ABC
must be the identity by P7.)

**Proposition 6.1**: P7 is equivalent to:

**P7'** : Let *l* and *l*' be two distinct lines in a projective plane with A,B,C on * l* and A',B',C' on
*l*'. Then there is one and only one projectivity of *l* onto* l*' such that ABC __^__ A'B'C'.

**P7'' **: Let *l* and *l*' be two distinct lines in a projective plane and X =* l.l'*. Then every
projectivity* l* __^__ *l*' sending X to itself is a perspectivity.

**Propositon 6.2** : P7 implies its dual statement:

**D7 **: Let P be a point in a projective plane. Let a, b, c and a', b',c' be two triples of lines
through P. Then there exists one and only one projectivity abc __^__ a'b'c'.

**Theorem 6.3**: P7 implies Desargues Theorem (P5).

#### 6.2 Geometry of Complex Numbers

**Def:** Let **C**_{\inf} denote the extended complex numbers **C** \cup {\inf}. Define for all a, b, c, d
such that ad - bc is not 0 a mapping f : **C**_{\inf} --> **C**_{\inf} by

ax + b
x --> --------,
cx + d

x --> \inf when x = \inf, c = 0 or x = -d/c, c \ne 0, and x --> a/c when x =\inf, c \ne 0.
f is called a *Möbius transformation*, or *fractional linear transformation*.
**Lemma 6.5**: The set of Möbius transformations form a group under composition; in
particular, each Möbius transformation is a one-to-one correspondence of **C**_{\inf} with itself.
Furthermore, a Möbius transformation is determined by its value on three elements of **C**_{\inf}.

**Theorem 6.6**: P7 holds in the real projective plane.

#### 6.3 Pappus' Theorem

**Theorem 6.7: (Pappus's Theorem)** Let *l* and *l*' be distinct lines in a projective plane with
P7. Let A, B, C be three distinct points on *l*, different from Y =* l.l*'. Let A', B', C' be three
distinct points on *l*', different from Y. Define P = AB'.A'B, Q = AC'.A'C, and R =
BC'.B'C. Then P, Q and R are collinear.

**Rmk**: This theorem says that P7 implies Pappus. The converse is also true, Pappus implies
P7, so they are equivalent and either could be taken as an axiom, which is why planes
satisfying P7 are called *Pappian* planes.

**N.B**. In Exercise 6.13, you prove that Pappus implies Desargue. Thus, Pappian planes are
always Desarguesian planes. However, the converse is not true in general. But, for finite
planes it is (**Wedderburn's Theorem** in algebra).