---
tag: BT5B00W898
kind: definition
name: Spherical geometry
msc: [51M10, 51M09]
defines: SQ9RQVANMG
uses: [T62369RFH6, 0S52TNNTH9, 0V3XAMF3C1]
sources:
  - url: https://en.wikipedia.org/wiki/Spherical_geometry
---

## Definition

Let $S^2 = \set{x \in \R^3 : \abs{x} = 1}$ be the unit [[sphere|T62369RFH6]], and let $x \cdot y$ be the [[dot product|0S52TNNTH9]] of $\R^3$.
*Spherical geometry* is the geometry on $S^2$ with these data:

1. The points are the points of $S^2$.
2. The lines are the *great circles*, the sets $S^2 \cap P$ where $P$ is a [[plane|0V3XAMF3C1]] through the origin.
3. The distance is $d(x, y) = \arccos(x \cdot y) \in [0, \pi]$, the length of a shortest arc of great circle from $x$ to $y$.
4. The angle between two great circles at a common point is the angle between their planes.

## Remarks

Example: the equator $\set{x_3 = 0} \cap S^2$ is a line, since the plane $x_3 = 0$ passes through the origin.
The points $N = (0, 0, 1)$, $A = (1, 0, 0)$, $B = (0, 1, 0)$ are pairwise at distance $\arccos 0 = \pi/2$, and the three planes $x_1 = 0$, $x_2 = 0$, $x_3 = 0$ are pairwise perpendicular, so the triangle $NAB$ has three right angles.
Non-example: the circle of latitude $\set{x_3 = 1/2} \cap S^2$ is not a line, because the plane $x_3 = 1/2$ does not pass through the origin.
Two distinct great circles always meet in a pair of antipodal points, so there are no parallels.
Two points that are not antipodal lie on exactly one great circle, but antipodal points lie on infinitely many.
On a sphere of radius $R$ the same definitions apply with distance $R \arccos(x \cdot y / R^2)$; the geometry has constant curvature $1/R^2$.