Hyperbolic Space

Hyperbolic Space

In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point. Parallel lines are not uniquely paired: given a line and a point not on that line, any number of lines can be drawn through the point which are coplanar with the first and do not intersect it. This contrasts with Euclidean geometry, where parallel lines are a unique pair, and spherical geometry, where parallel lines do not exist, as all lines (which are great circles) cross each other. Another distinctive property is the amount of space covered by the n-ball in hyperbolic n-space: it increases exponentially with respect to the radius of the ball, rather than polynomially.