The 3D fcc (face-centered-centered) lattice, which has the same packing ratio as the 3D hexagonal close packed lattice, has the following 12 vectors connecting each vertex with its neighbors:

$(1,-1,0), (-1,1,0), (-1,-1,0), (1,1,0)$

$(1,0,-1), (-1,0,1), (-1,0,-1), (1,0,1)$

$(0,1,-1), (0,-1,1), (0,-1,-1), (0,1,1)$

If I have an arbitrary point $(x,y,z)$, I need a piecewise formula for finding which vertex in the lattice is the closest to it. I do not want an "algorithmic" solution, i.e. calculating distances to many vertices or computing voronoi cells.

Thank You!

The 3D fcc (face-centered-cubic) lattice, which has the same packing ratio as the 3D hexagonal close packed lattice, has the following 12 vectors connecting each vertex with its neighbors:

$(1,-1,0), (-1,1,0), (-1,-1,0), (1,1,0)$

$(1,0,-1), (-1,0,1), (-1,0,-1), (1,0,1)$

$(0,1,-1), (0,-1,1), (0,-1,-1), (0,1,1)$

If I have an arbitrary point $(x,y,z)$, I need a piecewise formula for finding which vertex in the lattice is the closest to it. I do not want an "algorithmic" solution, i.e. calculating distances to many vertices or computing voronoi cells.

Thank You!