In the Wikipedia article on Cayley transform they provide an example for a map that sends an upper half-plane to the unit disk
$$f(z) = \frac{z – i}{z+i}$$
There is a depiction on the transform in the above image.
I want to animate the transformation, i.e. to produce a series of images there plane is transformed to the plane, while those colored lines (on the left) are gradually bent until they become circles (on the right).
I lack the knowledge on how to create a series of transformations. Does there exist a continuous function, that takes a real number $x \in [0, 1]$ and produces an automorphism $\varphi_x : \mathbb{C} \to \mathbb{C}$?
$$F(x) = \varphi_x$$
With a boundary condition
$$\varphi_0 = id$$
and
$$\varphi_1 = f$$
What is meant by continuous (except for visual perception)? Some continuity in $|| \cdot ||_{\infty}$ of functions?
Best Answer
The function \begin{eqnarray*} F(z,t) = \frac{z-it}{1+t(z+i-1)} \end{eqnarray*} satifies $F(z,0)=z$ and $F(z,1)= \frac{z-i}{z+i}$. Gives that a whirl ?