NonEuclidean

Non-Euclidean geometry refers to geometries that reject or modify Euclid's parallel postulate, which in standard Euclidean geometry asserts that through a point not on a given line there is exactly one line parallel to the given line. By altering this postulate, these geometries produce spaces with different notions of distance and angle. They are typically described as spaces of constant curvature, with Euclidean geometry arising as the zero-curvature case.

There are two classical branches: hyperbolic geometry (negative curvature) and elliptic geometry (positive curvature, often called spherical geometry when restricted to a two-dimensional surface). In hyperbolic geometry, through a point not on a line there are infinitely many lines that do not meet the given line; triangles have angle sums less than 180 degrees. Lines diverge, and parallelism behaves differently from Euclidean intuition. In elliptic geometry, through a point not on a line there are no parallel lines; every pair of lines intersects, and triangles have angle sums greater than 180 degrees. A standard model for elliptic geometry is the surface of a sphere, where geodesics are great circles.

Historically, non-Euclidean geometry emerged in the early 19th century as a challenge to Euclid's postulate. It was developed by Russian mathematicians Nikolai Lobachevsky and János Bolyai, with important contributions from Carl Friedrich Gauss. The term reflects a deliberate departure from Euclidean assumptions and established that coherent geometries could exist beyond the Euclidean framework.

In mathematics and physics, non-Euclidean geometry provides the language for curved spaces. In general relativity, spacetime is modeled as a curved manifold whose local geometry is non-Euclidean. It also informs models in cosmology, computer graphics, and navigation on curved surfaces.