---
tag: QH76Y4JD89
kind: definition
name: Elliptic geometry
msc: [51M10, 51M09]
defines: H5GB19W2H3
uses: [SQ9RQVANMG, T62369RFH6, 1DDYB6VVSR, 0S52TNNTH9]
sources:
  - url: https://en.wikipedia.org/wiki/Elliptic_geometry
---

## Definition

Let $n \ge 2$, let $S^n \subset \R^{n+1}$ be the unit [[sphere|T62369RFH6]], and let $x \cdot y$ be the [[dot product|0S52TNNTH9]].
*Elliptic $n$-space* is the [[real projective space|1DDYB6VVSR]] $\mathbb{RP}^n = S^n / \set{x \sim -x}$ with these data:

1. A point is a pair $\set{x, -x}$ of antipodal points of $S^n$, written $[x]$.
2. A line is the image of a great circle of $S^n$, the intersection of $S^n$ with a 2-dimensional linear subspace.
3. The distance is $d([x], [y]) = \arccos \abs{x \cdot y} \in [0, \pi/2]$.

*Elliptic geometry* is the geometry of these spaces; for $n = 2$ it is the geometry of the elliptic plane.

## Remarks

The distance is well defined, since replacing $x$ by $-x$ does not change $\abs{x \cdot y}$.
Example: two distinct great circles of $S^2$ meet in exactly one antipodal pair $\set{x, -x}$, so two distinct lines of the elliptic plane meet in exactly one point.
Two distinct points $[x] \ne [y]$ span a 2-dimensional subspace, so they lie on exactly one line.
Non-example: the sphere $S^2$ with great circles as lines, that is [[spherical geometry|SQ9RQVANMG]], does not satisfy the fact that two distinct points lie on exactly one line.
The antipodal points $(0, 0, 1)$ and $(0, 0, -1)$ lie on infinitely many great circles.
For this reason spherical geometry is sometimes called double elliptic geometry, and elliptic geometry single elliptic geometry.
The elliptic plane has constant curvature $1$, and the double cover $S^2 \to \mathbb{RP}^2$ is a local isometry.