Introduction to Non-Euclidean Geometry

This lecture introduces the concepts of non-Euclidean geometry, focusing on hyperbolic and spherical geometries as alternatives to Euclidean geometry. It covers the development of these geometries, the implications of negating the fifth postulate, and the significance of the 5th postulate in constructing these alternative geometries. The lecture explores the surfaces of hyperboloids, the concept of geodesic hyperbolas, and the projection of hyperboloids onto a disk. It also delves into the Tractricoid, a non-constructible curve, and the role of projective geometry in challenging the notion of parallels. The presentation emphasizes the importance of these alternative geometries in modern mathematics and their practical applications.