From the algebraic viewpoint, there is little difference between the description of the action of the complex projective group on complex projective space and the description of the action of the real projective group on real projective space.

Transformations of the form
are commonly called *Moebius transformations* or *linear fractional
transformations*.

- Show that there exists a unique transformation of the form
- Write the formula for the projective transformation that takes to (i.e., the inverse of the above transformation).
- Write the formula for the transformation that takes to .

As a trivial consequence of this exercise we have:

The preceding exercises -- exact copies of those of chapter 2 -- may lull the reader into believing that the geometry of the complex projective line is similar to that of the real projective line. We will presently see that the former is infinitely richer and more beautiful. For one thing, circles play a privileged role in complex projective geometry.

**Remark.**
From now on straight lines and circles will be simply called circles.
The intuitive idea is that a straight line is a circle of infinite
radius, or just the stereographic projection of a circle passing through
the north pole.

An obvious consequence of exercise 2.8 is the following important result.

To better understand Moebius transformations, let us consider three particular cases. The first two cases are already familiar: transformations of the form are translations of the number thought of as a vector on the plane, transformations of the form are dilations combined with rotations around the origin. The third case, the inversion , is more interesting.

The reader should always keep in mind that Moebius transformations are just complex projective transformations when seen in the model that identifies with plus a point at infinity. Changing from this model to the sphere usually leads to new insights such as the one in the exercise above. Another example is supplied by the following exercise:

Since we can multiply all entries of by the same nonzero complex number without changing the transformation, we may assume that the determinant of equals 1. Use this to identify the group as the double cover of the group of rotations .

- Let and be real numbers and let be a complex number. Show that the equation defines a circle if and only if . Conversely, show that any circle has an equation of this form.
- Use this to show that Moebius transformations send circles to circles.
- An
*anti-homography*is a transformation of the form , with . Show that anti-homographies also send circles to circles.

Besides sending circles to circles, Moebius transformations have the important property that they preserve angles and orientation. To place this in the proper context, we will study all transformations that satisfy this property.

A smooth map from an open subset of
to
that preserves angles
and orientations is said to be *conformal.*

- Show that a smooth map
from an open subset
of
to
is conformal if and only if the
*Cauchy-Riemann*equations hold:and - Let . Show that is conformal if and only if is identically zero.
- Show that if is a conformal transformation, then and
are harmonic functions:

- Show that the function is a conformal map.
- Show that the function (complex multiplication) is a conformal map. Conclude that any complex polynomial defines a conformal map.
- Assuming that is not identically zero, show that the function is conformal on its domain of definition. Conclude that a rational function (i.e., a quotient of two polynomials) is conformal on its domain of definition.

The previous exercise proves the following important result:

Using that Moebius transformations preserve angles, we can give a simple proof that the stereographic projection preserves angles.