Mapping Lines and Circles onto the Riemann Sphere

point

choice

line

circle

radius

angle

show lines

One of the great miracles of mathematics is the fact that an infinitely extended plane, which is densely packed with complex numbers, can be mapped onto a sphere with radius

1

2

and hence an area of

π

. Here you can see a decent fraction of the complex plane and the south pole of the Riemann sphere placed at the origin.

You can observe how straight lines or circles in the complex plane are transformed into circles on the sphere. The "radius" control changes the radius of the circle while "angle" alters the angle between the straight line and the real axis. "Point" is the center of the circle or one point of the line that you can move in the part of the complex plane shown.