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Zone diagram sweep

Each zone R_i is the set of points at least as close to its site p_i as to every other zone. The zones are grown from their sites and each boundary ray is fixed in increasing order of its distance to the site, like Dijkstra's algorithm. No pixel grid and no global iteration. The α slider moves continuously from the Voronoi diagram to the zone diagram.

Sites

From Voronoi cells to zones

α 1.00

Each zone competes with the other zones shrunk towards their sites, S_j = p_j + α (R_j − p_j). At α = 0 the competitors are the sites alone and the zones are the Voronoi cells.

Sweep level

all

At level s the picture shows R_i ∩ B(p_i, s). Everything below s is final: what lies further out never changes it.

Display

Colour each boundary arc by the zone it faces Contacts and corners Voronoi diagram Shrunk competitors αR_j Dependency chain under the pointer Rays per zone

Legend

arc of the bisector of p_i and a smooth part of R_j

straight segment facing a corner of R_j

contact: two zones touch while both grow

corner: the facing zone changes

Combinatorial skeleton

Boundary arcs of each zone in counterclockwise order, named by the zone they face.

Computation

The check measures max |ρ_i(u) − d(tip, ∪_j≠i R_j)| over the boundary. It is zero exactly for the zone diagram.