Geometry - Intuition and Concepts

What is geometry.- Parallelism: affine geometry.- From affine geometry to linear algebra.- Definition of affine space.- Parallelism and semiaffine mappings.- Parallel projections.- Affine coordinates and center of gravity.- Incidence: projective geometry.- Central perspective.- Far points and straight lines of projection.- Projective and affine space.-Semi-projective mappings and collineations.- Conic sections and quadrics; homogenization.- The theorems of Desargues and Brianchon.- Duality and polarity; Pascal's theorem.- The double ratio.- Distance: Euclidean geometry.- The Pythagorean theorem.- Isometries of Euclidean space.- Classification of isometries.- Platonic solids.- Symmetry groups of Platonic solids.- Finite rotation groups and crystal groups.- Metric properties of conic sections.- Curvature: differential geometry.- Smoothness.- Fundamental forms and curvatures.- Characterization of spheres and hyperplanes.- Orthogonal hyperface systems.- Angles: conformal geometry.- Conformal mappings.- Inversions.- Conformal and spherical mappings.- The stereographic projection.- The space of spheres.- Angular distance: spherical and hyperbolic geometry. The hyperbolic space. Distance on the sphere and in hyperbolic space. Models of hyperbolic geometry.- Exercises.- Solutions