Stereographic Projection - Other Formulations and Generalizations

Other Formulations and Generalizations

Some authors define stereographic projection from the north pole (0, 0, 1) onto the plane z = −1, which is tangent to the unit sphere at the south pole (0, 0, −1). The values X and Y produced by this projection are exactly twice those produced by the equatorial projection described in the preceding section. For example, this projection sends the equator to the circle of radius 2 centered at the origin. While the equatorial projection produces no infinitesimal area distortion along the equator, this pole-tangent projection instead produces no infinitesimal area distortion at the south pole.

In general, one can define a stereographic projection from any point Q on the sphere onto any plane E such that

• E is perpendicular to the diameter through Q, and
• E does not contain Q.

As long as E meets these conditions, then for any point P other than Q the line through P and Q meets E in exactly one point P′, which is defined to be the stereographic projection of P onto E.

All of the formulations of stereographic projection described thus far have the same essential properties. They are smooth bijections (diffeomorphisms) defined everywhere except at the projection point. They are conformal and not area-preserving.

More generally, stereographic projection may be applied to the n-sphere Sn in (n + 1)-dimensional Euclidean space En + 1. If Q is a point of Sn and E a hyperplane in En + 1, then the stereographic projection of a point P ∈ Sn − {Q} is the point P′ of intersection of the line with E.

Still more generally, suppose that S is a (nonsingular) quadric hypersurface in the projective space Pn + 1. By definition, S is the locus of zeros of a non-singular quadratic form f(x₀, ..., x_{n + 1}) in the homogeneous coordinates x_i. Fix any point Q on S and a hyperplane E in Pn + 1 not containing Q. Then the stereographic projection of a point P in S − {Q} is the unique point of intersection of with E. As before, the stereographic projection is conformal and invertible outside of a "small" set. The stereographic projection presents the quadric hypersurface as a rational hypersurface. This construction plays a role in algebraic geometry and conformal geometry.