# Mapping Circles by a Linear Fractional Transformation

Mapping Circles by a Linear Fractional Transformation

A linear fractional transformation (or Möbius transformation) in the complex plane is a conformal mapping that has the form , where , , , and are complex, with . The transformation transforms circles in the plane into circles in the plane, where straight lines can be considered to be circles of infinite radius.

w=f(z)=

az+b

cz+d

a

b

c

d

ad-bc≠0

f

z

w

In this Demonstration, the red circle is transformed into the blue circle of the form .

|z-z|=r

0

|w-w|=s

0