Visualizing conformal map from unit disk to upper half plane

Question

I understand that $$w=\frac{z-i}{z+i}$$ maps the open upper half plane $\mathbb{H}$ to the open unit disk $\mathbb{D}$, so $$z=i\frac{1+w}{1-w}$$ maps from $\mathbb{D}$ to $\mathbb{H}$. My question is how? Are there any good GIFs or visuals that could show which points map to where?

I am asking because ultimately I want to determine how the right half unit disk would map using this same map.

Answer 1

I have been looking for any GIF which illustrates this transformation, but haven't found any... So I decided to create my own. I made this GIF with a little python code. I have uploaded this code I just made here. I hope you like it!