Given two non-overlapping disks and , the loxodromic transformation is said to pair the two disks if (1) maps the circle that bounds to the circle that bounds , (2) maps the outside of to the inside of and the inside of to the outside of , (3) the source of lies inside and the sink lies inside , and (4) forward iterates of under shrink to smaller disks containing the sink whereas backward iterates of shrink to the source.¹

In following program, the large red disks centered on the negative real axis () and on the positive real axis () are paired. The slider n sets the number of forward and backward iterates of . Set n to its largest value to see the forward iterates of appear as lightening disks shrinking into the sink contained in , and its backward iterates appear as darkening disks shrinking into the source in .

You can drag the green disk to different locations and resize it with the radius control. Place the green disk so that it sits outside of both and , and use the m slider to increase the number of its iterates. Because lies outside , its forward iterates lie inside and converge to the sink of . Similarly, because lies outside , its backward iterates lie inside and converge to the source of .