Quartets on a Sphere

For any region on a sphere, it is impossible to construct a (plane) map of the region in which distances are faithfully represented. This fact can be proved by showing that any given region on a sphere contains a quartet (i.e., a four-point space)Four-point properties that is not isometrically embeddable in the plane. Actually, any region on a sphere contains a quartet whose six distances determine the radius of the sphere uniquely. Moreover, we present an equation to judge if a given quartet can be a quartet on a sphere of a suitable radius.