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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!

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  • $\begingroup$ (geometry) tag is deprecated, see the tag-info. Please, try to choose other suitable tag. $\endgroup$ Aug 28, 2017 at 14:15
  • $\begingroup$ An algorithmic solution is more easily expressible. If you are looking for a purely arithmetic formula without conditionals, you will have something that is very hard to implement. What are you going to do with such a formula if you get it? Gerhard "Thinks The Goal Is Wrong" Paseman, 2017.08.28. $\endgroup$ Aug 28, 2017 at 14:38
  • $\begingroup$ I'm fine with conditionals, which is why I said piecewise. I just want a simple method that does not require distance calculations to multiple points. $\endgroup$ Aug 28, 2017 at 15:20
  • $\begingroup$ Then start with the central cube. You will find that deciding on the orthant, followed by looking at (but not necessarily calculating) the Voronoi cells will lead to a simple algorithm based on relative size of coordinates. You can use translation to do this in other cubes. Gerhard "First, Make The Problem Local" Paseman, 2017.08.28. $\endgroup$ Aug 28, 2017 at 19:34

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Recall that the face-centred cubic lattice comprises all vectors in $\mathbb{Z}^3$ whose coordinate sum is even.

Let $(x, y, z) \in \mathbb{R}^3$. For each coordinate, define the discrepancy to be the distance to the closest integer, i.e. $\delta(x) := |x - \lfloor x \rceil |$.

For the two coordinates with lowest discrepancy, replace them with their nearest integers. For the coordinate with highest discrepancy, either replace it with the nearest integer or the second-nearest integer, depending on which one has the correct parity to make the coordinate sum even.

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