Although any stereographic projection misses one point on the sphere (the projection point), the entire sphere can be mapped using two projections from distinct projection points. In other words, the sphere can be covered by two stereographic parametrizations (the inverses of the projections) from the plane. The parametrizations can be chosen to induce the same orientation on the sphere. Together, they describe the sphere as an oriented surface (or two-dimensional manifold).
This construction has special significance in complex analysis. The point  in the real plane can be identified with the complex number . The stereographic projection from the north pole onto the equatorial plane is then
Similarly, letting  be another complex coordinate, the functions
define a stereographic projection from the south pole onto the equatorial plane. The transition maps between the - and -coordinates are then  and , with  approaching 0 as  goes to infinity, and vice versa. This facilitates an elegant and useful notion of infinity for the complex numbers and indeed an entire theory of meromorphic functions mapping to the Riemann sphere. The standard metric on the unit sphere agrees with the Fubini–Study metric on the Riemann sphere.