Circle Packings with Linear Fractional Transformations

with formula z↦

az+b

cz+d

a

b

c

d

range

1.

A linear fractional transformation of the complex plane has the form

z↦

az+b

cz+d

, where

a

,

b

,

c

, and

d

are complex, and

ad-bc≠0

. Such a map is conformal (i.e. it preserves angles locally) and preserves generalized circles (i.e. lines or circles map into lines or circles).

The transformation here acts on a hexagonal grid with disks inscribed in each cell. Mapping just three points, it is possible to calculate the radius and the coordinates of the centers of the new disks.