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).