Non Euclidean Space

In mathematics non euclidean geometry consists of two geometries based on axioms closely related to those that specify euclidean geometry.

Non euclidean space. Non euclidean geometry refers usually to geometrical spaces where the parallel postulate is false. Many scientific fields study data with an underlying structure that is a non euclidean space. Although the term is frequently used to refer only to hyperbolic geometry common usage includes those few geometries hyperbolic and spherical that differ from but are very close to euclidean geometry see table. Some examples include social networks in computational social sciences sensor networks in communications functional networks in brain imaging regulatory networks in genetics and meshed surfaces in computer graphics.

We may possibly find a coordinate system where we can do some of these but not all. One type of non euclidean geometry that is of interest is hyperbolic space also called negatively curved space. A space whose properties are based on a system of axioms other than the euclidean system. The geometries of non euclidean spaces are the non euclidean geometries.

Space curves inward in a curved non euclidean geometry we cannot find a set of coordinates which are mutually perpendicular where the coordinate lines are all parallel to each other and where each grid square has the same area. Even a two dimensional physical version of a hyperbolic space is impossible to make. Non euclidean geometry literally any geometry that is not the same as euclidean geometry. In the latter case one obtains hyperbolic geometry and elliptic geometry the traditional non euclidean geometries.

In three dimensions there are three classes of constant curvature geometries all are based on the first four of euclid s postulates but each uses its own version of the parallel postulate the flat geometry of everyday intuition is called euclidean geometry or parabolic geometry and the non euclidean geometries are called hyperbolic geometry or lobachevsky bolyai. As euclidean geometry lies at the intersection of metric geometry and affine geometry non euclidean geometry arises by either relaxing the metric requirement or replacing the parallel postulate with an alternative. When the metric requirement.