# hyperbolic geometry

A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), along with two diverging ultra-parallel lines.

Hyperbolic geometry is one of the two main types of non-Euclidean
geometry and the first to be discovered. It is concerned with saddle-surfaces,
which have negative curvature and on which
the geodesics are hyperbolas.
In hyperbolic geometry, contrary to the parallel
postulate, there exists a line *m* and a point *p* not on *m* such that at least two distinct lines parallel to *m* pass
through *p*. As a result: the sum of the angles of a triangle is less
than 180° and, for a right triangle, the square of the hypotenuse is
greater than the sum of the squares of the other two sides.