Circle Packings with Linear Fractional Transformations

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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).


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.


Contributed by: Enrique Zeleny (January 2015)
Open content licensed under CC BY-NC-SA



Snapshot 1 shows the hexagonal grid with disks.

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