Hyperbolic geometry
In Euclidean geometry, the angle of parallelism is a constant; that is, any distance  between parallel lines yields an angle of parallelism equal to 90°. In hyperbolic geometry, the angle of parallelism varies with the  function. This function, described by Nikolai Ivanovich Lobachevsky, produces a unique angle of parallelism for each distance . As the distance gets shorter,  approaches 90°, whereas with increasing distance  approaches 0°. Thus, as distances get smaller, the hyperbolic plane behaves more and more like Euclidean geometry. Indeed, on small scales compared to , where  is the (constant) Gaussian curvature of the plane, an observer would have a hard time determining whether he is in the Euclidean or the hyperbolic plane.