This text on geometry modeling is devoted to a number of central geometrical topics—
graphs of functions, transformations, (non-)Euclidean geometries, curves and surfaces—
and presents some elementary methods for analytical modeling and visualization
In 1872 F. Klein proposed his Erlangen Programme in which he suggested that different
geometries can be studied by the properties of the groups of transformations acting
on the geometry. The following geometries are represented in this way: Euclidean,
affine, projective, inversive, spherical, and hyperbolic.
B. Riemann’s (1868) idea was to represent geometries by ametric (differential) form
in a curvilinear coordinate system. The distance between points is measured as the minimum
length of curves (calculated using the metric form!) joining them. The intrinsic
geometry of a surface in space and Euclidean, spherical, and hyperbolic geometries are
represented in this way.