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), doi:10.1007/s00454-014-9627-0.
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.