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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.
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