  This paper is devoted to a coordinate-free approach to several classic
geometries such as hyperbolic (real, complex, quaternionic), elliptic
(spherical, Fubini-Study), and lorentzian (de Sitter, anti de Sitter) ones.
These geometries carry a certain simple structure that is in some sense
stronger than the riemannian structure. Their basic geometrical objects have
linear nature and provide natural compactifications of classic spaces. The
usual riemannian concepts are easily derivable from the strong structure and
thus gain their coordinate-free form. Many examples illustrate fruitful
features of the approach. The framework introduced here has already been shown
to be adequate for solving problems concerning particular classic spaces.
