Mapping Circles by a Linear Fractional Transformation

move

circle

1

2

a

b

c

d

z

0

r

show arrows

axes

f(z)

-0.5+(-0.1+1.)z

-0.1z-(0.5-0.3)

A linear fractional transformation (or Möbius transformation) in the complex plane is a conformal mapping that has the form

w=f(z)=

az+b

cz+d

, where

a

,

b

,

c

, and

d

are complex, with

ad-bc≠0

. The transformation

f

transforms circles in the

z

plane into circles in the

w

plane, where straight lines can be considered to be circles of infinite radius.

In this Demonstration, the red circle

|z-

z

0

|=r

is transformed into the blue circle of the form

|w-

w

0

|=s

.