This page displays a few families of conformal equivalences between subsets of $\mathbb{H}$ and $\mathbb{H}$, parametrized by half plane capacity (the animations themselves, however, are parametrized by the square root of the half plane capacity, since $\mathrm{hcap}(rA) = r^2\, \mathrm{hcap}(A)$).

# Removal of line segment

This family of conformal transformations is parametrized by the function $g_t(z) = \sqrt{z^2 + 4t},$ which is the unique conformal transformation that maps $\mathbb{H} \setminus [0, 2\sqrt{t}i]$ to $\mathbb{H}$ such that $|g_t(z) - z| \rightarrow 0\text{ as } z \rightarrow \infty.$ The line $[0, 2\sqrt{t}i]$ is displayed in black.

Note that in the square root, we pick the

# Removal of disk

This is the family of conformal transformations given by $g_t(z) = z + \frac{t}{z},$ which is the unique conformal transformation that maps $\mathbb{H} \setminus \sqrt{t}\mathbb{D}$ to $\mathbb{H}$ such that $|g(z) - z| \rightarrow 0\text{ as } z \rightarrow \infty.$ The region $\sqrt{t} \mathbb{D}$ is colored in green.