Voronoi_cell

Voronoi cell

A Voronoi cell is the locus of all points that are closer to a particular point than to any other of a finite collection of such points. (See the Wikipedia entry).

FLUX considers space to be divided into a 3-D quasi-Voronoi foam of cells that are closer to particular fluxons than to any other. The network of cells is a quasi-Voronoi foam, and not a Voronoi foam, because the core of each cell is not a single point but rather a line segment (one fluxel of the many that compose a fluxon). The magnetic flux associated with each fluxel is considered to be able to expand perpendicular to the direction of the fluxel, to fill the quasi-Voronoi cell of the fluxel.

Because 3-D Voronoi foams are hard to determine computationally, and because quasi-Voronoi foams are even more difficult to determine (the bondaries of a quasi-Voronoi foam surrounding a collection of line segments are determined by bivariate fourth-order polynomials), FLUX approximates the 3-D problem with a simpler 2-D one: each fluxel's cell is approximated with a prism that is found by calculating a Voronoi cell in the 2-D cross-sectional plane of the fluxel, and then extruded along the length of the fluxel.

The fluxel of interest and all neighboring fluxels are projected onto the perpendicular plane as points. The projection is non-Euclidean: fluxels that are out of the perpendicular plane of the subject fluxel are moved artificially far away from the 2-D origin.