COURSE AIMS AND OBJECTIVES:
The aim of the course is to introduce students to the fundamental concepts and results of two standard non-Euclidean geometries (spherical and hyperbolic), as well as some "hypergeometries" (affine and projective). Geometric content is developed by analogy, starting from the model of Euclidean plane and its isometry group, to the isometry groups of other geometries for the corresponding models. The spherical model serves to build spherical and projective geometry, and the model of a two-sheet hyperboloid in 3-dimensional Lorentz-Minkowski space, as well as the Klein model of the open circle derived from it, to build hyperbolic geometry.
COURSE DESCRIPTION AND SYLLABUS:
1. Euclidean plane in the model of R^2 (analytical approach). Points and lines, incidence relation.
2. Mutual positions of two lines, distance of two points, pencil of lines of Euclidean plane.
3. Group of isometries of Euclidean plane.
4. Affine geometry. Group of isometries.
5. Geometry on the sphere on the model of S^2 in R^3. Points and lines, incidence relation.
6. Mutual positions of two lines, distance of two points, pencil of lines.
7. Group of isometries of spherical geometry.
8. Spherical trigonometry.
9. Projective plane. Homogeneous coordinates.
10. Desargues and Pappus theorem.
11. Projective group. Elliptic plane.
12. Hyperbolic plane in the model of a two-sheet hyperboloid H^2 in Lorentz-Minkowski space R^3_1. Points and lines, incidence relation.
13. Mutual positions of two lines, distance of two points, pencil of lines (especially parallel and ultraparallel lines).
14. Group of isometries of hyperbolic geometry.
15. Hyperbolic trigonometry.
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Euclidean and non-Euclidean Geometry - an Analytic Approach, P. J. Ryan, Cambridge University Press, 1991.
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O euklidskoj i neeuklidskim geometrijama, A. I. Fetisov, Školska knjiga, Zagreb, 1981.
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