Elliptic geometry

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, 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.