Linear Fractional Transformations

The function is
w = az + b
cz + d
The red grid is in the z = x+iy domain, the blue image is in the w = u+iv domain. These transformations are also called Möbius functions or bilinear transformations.

We begin by considering
w = (2+i) z + (2-i)
(1-i) z + (4+i)



Along the real axis, replace z by t.


Along the imaginary axis, replace z by it.

We see linear motion turned circular. If one considers the point at infinity, then lines are large circles. We can show that a linear fractional transformation takes circles to circles (counting lines as circles through infinity).

[Maple Plot]

Notice how the radial lines transform to great circles. Consider
w = (2+i) z + (2-i)
(1-i) z + (4+i)
Let a domain grid (blue) move around the unit circle and watch the image (red).