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...Notes on non euclidean geometries gabor moussong budapest semesters in mathematics contents introduction part one warmup ane geometry spherical two inversive inversion plane m obius transformations and complex numbers three projective space incidence coordinates cross ratio of the line conics four models hyperbolic model poincar e hyperboloid equivalence five parallelism trigonometry applications some special questions what is very general vague terms study shapes with mathematical rigor it a broad subject within framework semester course we can only touch upon selected topics present these from specic point view goals this to show how nds its place among manyabstract structures modern shall treat through powerful methods other chapters abstract linear algebra calculus groups opens up several channels which classical linked today s research advanced notably dierential topology group theory history has produced many dierent geometric systems oldest familiar typically such system or foll...