# Circle Packings with Linear Fractional Transformations

Circle Packings with Linear Fractional Transformations

A linear fractional transformation of the complex plane has the form , where , , , and are complex, and . 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).

z↦

az+b

cz+d

a

b

c

d

ad-bc≠0

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.