Examples
Example : Scale and rotate
Let \(T(z)=az\). If \(a=r\) is real this scales the plane. If \(a=e^{iθ}\) it rotates the plane. If \(a=re^{iθ}\) it does both at once.
Note that \(T\) is the fractional linear transformation with coefficients
Example : Scale and rotate and translate
Let \(T(z)=az+b\). Adding the \(b\) term introduces a translation to the previous example.
Note that \(T\) is the fractional linear transformation with coefficients
Example : Inversion
Let \(T(z)=\frac{1}{z}\). This is called inversion. It turns the unit circle inside out. Note that \(T(0)=∞\) and \(T(∞)=0\). In the figure below the circle that is outside the unit circle in the \(z\) plane is inside the unit circle in the w plane and vise-versa. Note that the arrows on the curves are reversed.
Note that \(T\) is the fractional linear transformation with coefficients
