A Combinatorial Coordinate System for the Slanted Cubic Grid

When the cubic grid is compressed in a diagonal direction, cubes become nonregular truncated octahedra and the cubic grid becomes combinatorially equivalent to the body centered cubic (BCC) grid. We propose a combinatorial coordinate system for this grid, which addresses all the cells in the grid through integer triplets and which reflects the grid geometry. This addressing scheme, together with the fact that each object in the BCC grid is manifold, simplifies topological analysis and processing of digital objects. For illustration purposes, we present a naive algorithm for finding the closure of a set of cells in the slanted cubic grid, which relies heavily on the use of combinatorial coordinates.