Two-Dimensional Geometries

C. Herbert Clemens, Ohio State University, Columbus, OH.

Introduction Neutral geometry: Euclid's postulates for plane geometry Euclidean (plane) geometry: Rectangles and cartesian coordinates Concurrence and circles in Euclidean geometry Spherical geometry: Surface area and volume of the $R$-sphere in Euclidean 3-space Usual dot-product for three-dimensional Euclidean space: Euclidean three-space as a metric space Transformations $K$-geometry: Changing coordinates Uniform coordinates for the two-dimensional geometries Return to spherical geometry: Spherical geometry from an advanced viewpoint Hyperbolic geometry: The curvature $K$ becomes negative Definitions Bibliography Index