flooru

floorU is a generalized floor operator defined with respect to a positive unit size U. For any real number x, floorU(x) returns the greatest multiple of U that is less than or equal to x. Equivalently, floorU(x) = floor(x / U) · U. In higher dimensions, this operation is applied component-wise to vectors, producing a grid-aligned coordinate where each component lies on a U-sized cell boundary.

This generalization unifies the standard floor function with grid discretization. When U = 1, floorU coincides with the ordinary floor function. The operator is monotone (if a ≤ b then floorU(a) ≤ floorU(b)) and idempotent (floorU(floorU(x)) = floorU(x)). It maps real-valued inputs to the set of multiples of U, effectively snapping values to the left/up grid line of the corresponding cell.

In practice, floorU is used to discretize continuous spaces, particularly in computational settings. It supports spatial indexing, voxelization, and grid-based simulations by converting coordinates or measurements to fixed-grid coordinates. It also provides a simple mechanism for aligning measurements to a common resolution, which is useful in data aggregation and visualization.

Examples illustrate its behavior: with U = 0.5, floorU(1.2) equals 1.0 and floorU(1.0) equals 1.0; with U = 1, floorU(x) reduces to the ordinary floor function floor(x). Variants include using floorU for indexing (producing integer grid indices) or for coordinate snapping (producing grid-aligned coordinates).