Spherical geometry

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 $1/R^2$.