The RectangularLattice is a helper class that maps all of N-D space into a regular, rectangular grid of cells identified by integer coordinates. The grid is defined by an origin point and a vector indicating spacing in each dimension.
The figure shows an example RectangularLattice in 2D, with its origin (circled) at (-0.6, -0.2) and spacing (1.2, 0.8). Given a query point, the RectangularLattice will return the coordinates of the cell that contains the point. It will also return the bounding box of a cell, or the coordinates of a cell's lower-left corner.
The following example shows the use of the RectangularLattice. First, include the header and (if desired) declare type aliases. Using const int in2d = 2 makes a 2D lattice.
../../examples/spin_introduction.cpp
Use the RectangularLattice to find grid cells.
../../examples/spin_introduction.cpp
The Mortonizer (along with its associated class MortonBase) implements the Morton index, an operation that associates each point in N-D space with a point on a space-filling curve1. The PointHash class adapts the Mortonizer to provide a hashing functionality for use with std::unordered_map or similar container classes.
The math of the Morton index works with integers. Thus the Mortonizer and its dependent class PointHash will not work with floating point coordinates. The following code example shows how the cells of a RectangularLattice, which have integer coordinates, can be used with a hash table.
The Mortonizer works by interleaving bits from each coordinate of the input point and finite computer hardware limits its resolution. If the MortonIndexType is 64-bits, then in 2D it can uniquely index the least significant 32 bits of the x- and y-coordinates. In 3D, it can uniquely index the least significant 21 bits of the x-, y-, and z-coordinates.
To use the PointHash, include the header and (as desired) declare type aliases.
../../examples/spin_introduction.cpp
The RectangularLattice grid cell associated with a query point can be stored, using a PointHash, in a std::unordered_map.
../../examples/spin_introduction.cpp
Footnotes
The Morton index is also known, among other things, as the Z-order curve: see its Wikipedia page.↩