This page displays a few families of conformal equivalences between subsets of and , parametrized by half plane capacity (the animations themselves, however, are parametrized by the square root of the half plane capacity, since ).
Removal of line segment
This family of conformal transformations is parametrized by the function which is the unique conformal transformation that maps to such that The line 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 which is the unique conformal transformation that maps to such that The region is colored in green.