This eventually led to the concept of the extended complex plane.
2.
However, what happens in the case of the extended complex plane?
3.
This topological space, the complex plane plus the point at infinity, is known as the extended complex plane.
4.
Intuitively, it is almost as if the convergent continued fraction maps the entire extended complex plane into a single point.
5.
The group " G " " c " acts by M�bius transformations on the extended complex plane.
6.
If " f " extends to a quasiconformal mapping of the extended complex plane then " ? is by definition a quasicircle.
7.
When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere.
8.
Note that, because of the assumptions on the Beltrami coefficient, a M�bius transformation of the extended complex plane can be applied to make the singular set of the Beltrami coefficient compact.
9.
There are two points at infinity ( positive, and negative ) on the real number line, but there is only one point at infinity ( the north pole ) in the extended complex plane.
10.
For example, projective geometry is about the group of projective transformations that act on the real projective plane, whereas inversive geometry is concerned with the group of inversive transformations acting on the extended complex plane.
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