Non-Euclidean Geometries

  • Instructor: dr Gábor Moussong
  • Contact: gabor.moussong at ttk dot elte dot hu
  • Prerequisites: Standard first-year linear algebra (operations on vectors, abstract vector spaces, linear transformations and matrices, quadratic forms). Some fundamental concepts in group theory (homomorphisms, subgroups and normal subgroups) are also indispensable. Naturally, some familiarity with standard facts of (high school level) Euclidean geometry is useful.
  • Text: G. Moussong, Notes on Non-Euclidean Geometries
Course description:

Beginner's introduction to various types of non-Euclidean geometry through concrete models (no axioms). Emphasis is on the role of transformations, and on how the technique of modern mathematics is utilized to get comprehensive understanding of different classical geometric systems.


  • Affine geometry: affine transformations of the plane, affine invariants, linear algebra and affine geometry. Orthogonal matrices and Euclidean geometry.
  • Spherical geometry: spherical triangles and trigonometry, structure of orthogonal groups.
  • Inversive geometry: inversions, Möbius transformations, and their invariants, Poincaré extension.
  • Projective geometry: the projective plane, homogeneous coordinates, projective transformations, cross ratio, conics, polarity.
  • Hyperbolic geometry: Projective, conformal, and quadratic form models of hyperbolic plane. Transformations, distance, angle, area. Some formulas of hyperbolic trigonometry.