of the sphere (minus the north pole ) to the plane by sending to the intersection of the line with the plane. Stereographic projection of a 3-sphere (again removing the north pole) maps to three-space in the same manner. (Notice that, since stereographic projection is conformal, round spheres are sent to round spheres or to planes.) A somewhat different way to think of the one-point compactification is via the exponential map. Returning to our picture of the unit two-sphere sitting on the Euclidean plane: Consider a geodesic in the plane, based at the origin, and map this to a geodesic in the two-sphere of the same length, based at the south pole. Under this map all points of the circle of radius are sent to the north pole. Since the open unit disk is homeomorphic to the Euclidean plane, this is again a one-point compactification. The exponential map for 3-sphere is similarly constructed; it may also be discussed using the fact that the 3-sphere is the Lie group of unit quaternions. Coordinate systems on the 3-sphere The four Euclidean coordinates for are redundant since they are subject to the condition that . As a 3-dimensional manifold one should be able to parameterize by three coordinates, just as one can parameterize the 2-sphere using two coordinates (such as latitude and longitude). Due to the nontrivial topology of it is impossible to find a single set of coordinates that cover the entire space. Just as on the 2-sphere, one must use at least two coordinate charts. Some different choices of coordinates are given below. Hyperspherical coordinates It is convenient to have some sort of hyperspherical coordinates on in analogy to the usual spherical coordinates on . One such choice — by no means unique — is to use , where where and run over the range 0 to , and runs over 0 to 2. Note that, for any fixed value of , and parameterize a 2-sphere of radius , except for the degenerate cases, when equals 0 or , in which case they describe a point. The round metric on the 3-sphere in these coordinates is given by and the volume form by These coordinates have an elegant description in terms of quaternions. Any unit quaternion can be written as a versor: where is a unit imaginary quaternion; that is, a quaternion that satisfies . This is the quaternionic analogue of