
Hans Havlicek: Projective Geometry
This series of lectures for students of Descriptive Geometry (teacher's programme) gives an introduction to projective, affine and Euclidean geometry.


Contents
The subsequent list refers to my lectures in the years 20012003 according to the old curriculum.

Projective plane with 13 points
Axiom of Desargues

1. Affine and Projective Planes
 Axioms of affine and projective planes
 Connection between affine and projective planes
 Principle of duality
2. Projectivities and Collineations
 Perspectivities
 Projectivities
 Axioms of Desargues and Pappos
 Hessenberg's theorem
 Perspective and projective collineations
 Harmonic tetrads
 Fano's axiom
3. Affinities
 Perspectivities in affine planes
 Perspective and projective affinities
4. Conics
 Steiner's definition
 Pascal's theorem
 Projectivities on conics
 External and internal points
 Nuclei
 Polarity with respect to a conic
 Higherorder contact of conics
 Pencils of conics
 Conics in affine planes
5. Affine and Projective Spaces
 Axioms of affine and projective spaces
 Connection between affine and projective spaces
6. Properties of Projective Spaces
 Lattice of subspaces
 Dimension formula of Grassmann
7. Properties of Affine Spaces
 Parallel subspaces
 Hyperplane at infinity
 Affine geometry in projective terms
8. The Dual of a Projective Space
 Pencils of hyperplanes
 Dual space
 Principle of duality
 Bidual of a projective space
9. Projectivities and Collineations of Projective Spaces
 Projectivities
 Perspective and projective collineations
 Transitivity properties of the projective group

Dual polyhedra

10. Correlations
 Correlations
 Polarities
 Conjugate points
 Autopolar simplices
 absolute points
 Types of polarities
11. Quadrics
 Quadratic sets
 Quadrics
 Polarity with respect to a quadric
 Common points of two conics
 Reguli
 Quadrics in affine spaces

General linear complex of lines

12. Null Polarities and Line Geometry in 3Spaces
 Null polarities in 3spaces
 General and special linear complexes of lines
 Hyperbolic, parabolic, and elliptic congruences of lines
13. Twisted Cubics
 Seydewitz's definition
 Chords
 tangents and osculating planes
 Polarity with respect to a twisted cubic
 Cubic developables
 Axes


14. Projective and Affine Spaces over Vector Spaces
 Projective spaces over vector spaces
 Fundamental theorem of projective geometry
 Affine spaces over vector spaces
 Fundamental theorem of affine geometry
 Representation of collineations and affinities in terms of semilinear mappings
 Projective and affine coordinates
 Cross ratios and affine ratios
15. Real and Complex Projective Spaces
 Separating quadruplets
 Complexification of real spaces
 Complex conjugate elements
16. Euclidean Spaces
 Orthogonality in terms of an absolute polarity
 Similarities
 Spheres
 Direct and opposite congruence transformations
 Euclidean spaces over vector spaces
 Angles and lengths
 Axes and umbilical points of quadrics
 Quadrics of revolution
 Foci of conics
