Projective Geometry I - Lecture Notes

5.1 Fano's Axiom P6

Def: complete quadrangle, diagonal points.

P6: (Fano's Axiom) The diagonal points of a complete quadrangle are not collinear.

Proposition 5.2: The real projective plane satisfies P6.

Temporary definition: a Fano Plane

Discuss why this is temporary.

Def: complete quadrilateral, diagonal lines.

Prop 5.4: If \pi is a Fano plane, then so is its dual \pi*.

5.2 Harmonic Points

Def: An ordered quadruple of distinct points A, B, C, D on a line is called a harmonic quadruple if there is a complete quadrangle XYZW such that A and B are diagonal points of the quadrangle and C and D are on the remaining two sides. H(A,B; C,D).

Prop 5.7: Let A,B,C be three distinct points on a line. Then (assuming P6) there is a point D such that H(A,B; C,D). Furthermore, if P5 is assumed, this point is unique. (D is called the 4th harmonic point of A, B, C, or the harmonic conjugate of C wrt A and B.)

Prop 5.8: Let H(A,B; C,D). Then (assuming P5) H(C,D; A,B).

Example: In PG(2,3) there are only 4 points on a line, they always form a harmonic quadruple in any order.

(note the slighting of nondesarguesian planes in the comment after this example).

Example: Cross ratio in the Euclidean plane.

Example: Cross ratio for lines in the Euclidean plane.

5.3 Perspectivities and Projectivities

Def: perspectivity, projectivity

Prop 5.14: Let l be a line. Then the set of projectivities of l into itself forms a group, which we will call PJ(l).

Prop 5.15: PJ(l) is 3-transitive.

Lemma 5.17: Suppose that ABCD ^ A'BC'D' are perspective in a projective plane satisfying P5 and P6. If H(A,B;C,D) then H(A',B; C',D').

Prop. 5.18: Assuming P5 and P6, a projectivity sends harmonic points into harmonic points.

Thus PJ(l) is not 4-transitive in Desarguesian planes.