# Elliptic geometry

**Elliptic geometry** (sometimes known as **Riemannian geometry**) is a non-Euclidean geometry, in which, given a line **L** and a point **p** outside **L**, there exists no line parallel to **L** passing through **p**.

Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which asserts that there is exactly one line parallel to *L* passing through *p*. In elliptic geometry, there are no parallel lines at all. Elliptic geometry has other unusual properties. For example, the sum of the angles of any triangle is always greater than 180°.

The simplest model of elliptic geometry is that of spherical geometry, where *points* are points on the sphere, and *lines* are great circles through those points. On the sphere, such as the surface of the Earth, it is easy to give an example of a *triangle* that requires more than 180°: For two of the sides, take lines of longitude that differ by 90°. These form an angle of 90° at the North Pole. For the third side, take the equator. The angle of any longitude line makes with the equator is again 90°. This gives us a triangle with an angle sum of 270°, which would be impossible in Euclidean geometry.

Elliptic geometry is sometimes called Riemannian geometry, in honor of Bernhard Riemann, but this term is usually used for a vast generalization of elliptic geometry.