Hyperbolic geometry is used very often in space, such as space travel and gravitational pulls and rotations of planets. This geometry is used most often in space because of Einstein's general Theory of Relativity assumes that space is not a Euclidean space, but a hyperbolic one.
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The 2 types of non-Euclidean geometries are hyperbolic geometry and ellptic geometry.
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A Russian mathematician named Nikolai Ivanovich Lobachevsky is the man credited with inventing hyperbolic geometry. Nikolai lived from 1792 to 1856.
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Answer The two commonly mentioned non-Euclidean geometries are hyperbolic geometry and elliptic geometry. If one takes "non-Euclidean geometry" to mean a geometry satisfying all of Euclid's postulates but the parallel postulate, these are the two possible geometries.