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 non-negative real axis to the branch cut, and so the function is not defined on $z = iy$ for $y \lt 2\sqrt{t} i$. In particular, the conformal map does not extend to the origin, hence we see that the map is not conformal "at the origin". Instead, it flattens out the right angle to a horizontal line.

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.