Non Euclidean Space

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.

Non euclidean space. A space whose properties are based on a system of axioms other than the euclidean system. They include elliptic geometry where the sum of the angles of a triangle is more than 180 and hyperbolic geometry where this sum is less than 180. 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.

Non euclidean geometry refers usually to geometrical spaces where the parallel postulate is false. In the latter case one obtains hyperbolic geometry and elliptic geometry the traditional non euclidean geometries. One type of non euclidean geometry that is of interest is hyperbolic space also called negatively curved space.

Starting with the notion of hyperbolic representations for hierarchical data two years ago a major push has resulted in new ideas for representations in non euclidean spaces new algorithms and models with non euclidean data and operations and new perspectives on the underlying functionality of non euclidean ml. In mathematics non euclidean geometry consists of two geometries based on axioms closely related to those that specify euclidean geometry. Depending on the specific axioms from which the non euclidean geometries are developed in non euclidean spaces the latter may be classified in accordance with various criteria.

When the metric requirement. 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. We may possibly find a coordinate system where we can do some of these but not all.

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. Many scientific fields study data with an underlying structure that is a non euclidean space. 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.