Spherical Geometry - Core Definitions

Spherical geometry studies the surface of a sphere, a non-Euclidean geometry with constant positive curvature. Unlike Euclidean geometry, all lines (great circles) eventually intersect, and there are no parallel lines.

DefinitionSpherical Space

The 2-sphere $S^2$ of radius $R$ has metric inherited from Euclidean 3-space:

ds^2 = R^2(d\theta^2 + \sin^2\theta \, d\phi^2)

in spherical coordinates $(\theta, \phi)$ . Lines are great circles (circles of maximum radius, lying in planes through the center).

Triangles on a sphere have angle sum exceeding $\pi$ . The spherical excess $E = (\alpha + \beta + \gamma) - \pi$ equals the area divided by $R^2$ :

A = R^2 E = R^2(\alpha + \beta + \gamma - \pi)

This contrasts with hyperbolic geometry (angle defect) and Euclidean geometry (angle sum exactly $\pi$ ).

DefinitionSpherical Distance

The distance between points $p$ and $q$ on the unit sphere is the angle (in radians) subtended at the center:

d(p,q) = \cos^{-1}(p \cdot q)

where $p, q \in \mathbb{R}^3$ are unit vectors. Great circle arcs minimize this distance.

ExampleLongitude and Latitude

On Earth, meridians (lines of constant longitude) are great circles, while parallels (lines of constant latitude) are generally not great circles (except the equator). The shortest path between two cities follows a great circle route, often surprisingly different from the constant-latitude path.

Remark

Historical significance: spherical geometry was essential for navigation and astronomy. Ancient Greek astronomers (Hipparchus, Ptolemy) developed spherical trigonometry for celestial calculations. Modern GPS systems use spherical (actually ellipsoidal) geometry for positioning.

The isometry group of $S^2$ is $O(3)$ (orthogonal transformations), with the orientation-preserving subgroup $SO(3)$ acting as rotations. Spherical geometry has constant curvature $K = 1/R^2$ .