Lectures on Projective Geometry

EEFN (First Frensh-Nordic Summerschool in Geometry)




The course in projective geometry will cover both the basic theory and some applications. We will start with the definition and properties of projective spaces. Special emphasis will be on the connection between projective, affine and Euclidean spaces; the so called stratified approach. Another aspect of special important to applications is projections from a higher dimensional projective space to another with lower dimension. We will also introduce some classical construction problems and invariant theory. One application that we will look into in a little bit more detail is multiple view geometry, a topic arising from trying to reconstruct a rigid three-dimensional world from a number of its two- dimensional images. We will introduce some tensor formalism to describe the relations between corresponding point in different images. In this case the projections are from 3D to 2D or sometimes from 2D to 1D.


  1. To understand and being able to use the basic concepts of projective geometry
  2. Knowledge of synthetic projective geometry
  3. To understand and being able to use coordinate based projective geometry (homogeneous coordinates)
  4. To understand the concept of multiple view geometry in computer vision


This tutorial focuses on the understanding and use of multiple view tensors in computer vision. We will cover the following topics:
  1. Lecture 1: Projective geometry I (introduction+synthetic)
  2. Lecture 2: Projective geometry II (synthetic)
  3. Lecture 3: Projective geometry III (analytic)
  4. Lecture 4: Projective geometry IV (analytic+stratification)
  5. Lecture 5: Cartesian tensors
  6. Lecture 6: Tensors and projective geometry
  7. Lecture 7: Multiple view geometry in computer vision
  8. Lecture 8: Auto-calibration
  9. References:
Lecture manuscript will be handed out during the course.

My home page

My ECCV'98 paper about multiple view tensors:
A Common Framework for Multiple-View Tensors

Anders Heyden
Department of Mathematics (LTH)
Lund Institute of Technology / Lund University
P.O. Box 118, S-221 00 LUND

Room: 453A
Direct Phone: +46 46 22 204 91
Dept. Phone: +46 46 22 285 37
Fax: +46 46 22 240 10
e-mail: Anders.Heyden@math.lth.se

Last edited, 1999-03-05,