Non-Euclidean Geometries

This lecture introduces various non-Euclidean geometries, including the hyperbolic geometry associated with the Poincaré disk and the intriguing tractricoid model. The tractricoid, a surface of revolution, is explored for its properties akin to those of a sphere. The lecture delves into the concept of geodesics projection from the Poincaré disk to the hyperboloid. It also discusses the history of non-Euclidean geometries, the tractrix curve, and the pseudosphere. The presentation covers the importance of the fifth postulate in different geometries and the modern reformulation of geometric postulates by Hilbert. Additionally, it touches upon the projective geometry, which challenges the Euclidean foundations and was invented by Desargues, an architect, for practical applications like perspective drawing and stereotomy.