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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.


The subsequent list refers to my lectures in the years 2001-2003 according to the old curriculum.


Projective plane with 13 points

Axiom of Desargues
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
  • Higher-order 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
Dual polyhedra

10. Correlations

  • Correlations
  • Polarities
  • Conjugate points
  • Auto-polar 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
General linear complex of lines

12. Null Polarities and Line Geometry in 3-Spaces

  • Null polarities in 3-spaces
  • 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

Quick Links

H. Havlicek: Lineare Algebra für Technische Mathematiker
(in German)

H. Havlicek: Multilinear Algebra
H. Havlicek: Evaluation of Teaching

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Last modified on July 29th, 2007.