A Möbius-invariant power diagram and its applications to soap
bubbles and planar Lombardi drawing.
D. Eppstein.
Invited talk at EuroGIGA Midterm
Conference, Prague, Czech Republic, 2012.
Discrete
Comput. Geom. 52 (3): 515–550, 2014 (Special issue for SoCG 2013).
This talk and journal paper combines the results from "Planar Lombardi drawings for subcubic graphs" and "The graphs of planar soap bubbles". It uses three-dimensional hyperbolic geometry to define a partition of the plane into cells with circular-arc boundaries, given an input consisting of (possibly overlapping) circular disks and disk complements, which remains invariant under Möbius transformations of the input. We use this construction as a tool to construct planar Lombardi drawings of all 3-regular planar graphs; these are graph drawings in which the edges are represented by circular arcs meeting at equal angles at each vertex. We also use it to characterize the graphs of two-dimensional soap bubble clusters as being exactly the 2-vertex-connected 3-regular planar graphs.