Riemann Sphere

a geometric representation introduced by B. Riemann for the set of complex numbers. The complex number

z = x + iy = r(cos ɸ + i sin ɸ) = re^iɸ

can be represented by the point in the complex plane with the Cartesian coordinates x,y or the polar coordinates r, ɸ. To construct a Riemann sphere, a sphere is placed so as to be tangent to the complex plane at the origin of coordinates, and the points of the plane are mapped onto the surface of the sphere by means of a stereographic projection.

Each complex number is thus represented by a corresponding point of the sphere, which is called a Riemann sphere. The number 0 here is represented by the south pole of the Riemann sphere. Numbers with the same argument ɸ = constant (rays in the complex plane) are represented on the sphere by meridians. Numbers with the same modulus r = constant (circles in the complex plane) are represented by parallels of the Riemann sphere. No point of the complex plane corresponds to the north pole of the Riemann sphere. To preserve the one-to-one correspondence between points of the complex plane and the Riemann sphere, there is adjoined to the plane a “point at infinity,” which is assumed to correspond to the north pole and designated by z = ∞. Thus, the complex plane contains a single point at infinity, in contrast to a projective plane.

Suppose a system of rectangular Cartesian coordinates ξ, n, ζ is introduced in such a way that the ξ-axis and η-axis coincide with the x-axis and y-axis, respectively. There then corresponds to the point x + iy of the complex plane the point

of the Riemann sphere. The equation of the sphere is ξ² + η² + ζ² - 2Rζ = 0.