This is a module to convert between one dimensional distance along a
Hilbert curve, h
, and N-dimensional coordinates,
(x_0, x_1, ... x_N-1)
. The two important parameters are N
(the number of dimensions, must be > 0) and p
(the number of
iterations used in constructing the Hilbert curve, must be > 0).
We consider an N-dimensional hypercube of side length 2^p
.
This hypercube contains 2^{N p}
unit hypercubes (2^p
along
each dimension). The number of unit hypercubes determine the possible
discrete distances along the Hilbert curve (indexed from 0
to
2^{N p} - 1
). The image below illustrates the situation for
N=2
and p=3
.
This is the third iteration (p=3
) of the Hilbert curve in two
(N=2
) dimensions. Distances, h
, along the curve are
labeled from 0 to 63 (i.e. from 0 to 2^{N p}-1
). The provided
functions translate between N-dimensional coordinates and the one
dimensional distance. For example, between (x_0=4, x_1=6
) and
h=36
.
An animation of the same case in 3-D is available on YouTube. To watch the video, click the link below. Once the YouTube video loads, you can right click on it and turn "Loop" on to watch the curve rotate continuously.
This module is based on the C code provided in the 2004 article "Programming the Hilbert Curve" by John Skilling,
I was also helped by the discussion in the following stackoverflow post,
which points out a typo in the source code of the paper. The Skilling code
provides two functions TransposetoAxes
and AxestoTranspose
. In this
case, Transpose refers to a specific packing of the integer that represents
distance along the Hilbert curve (see below for details) and
Axes refer to the N-dimensional coordinates. Below is an excerpt from the
documentation of Skilling's code,
//+++++++++++++++++++++++++++ PUBLIC-DOMAIN SOFTWARE ++++++++++++++++++++++++++
// Functions: TransposetoAxes AxestoTranspose
// Purpose: Transform in-place between Hilbert transpose and geometrical axes
// Example: b=5 bits for each of n=3 coordinates.
// 15-bit Hilbert integer = A B C D E F G H I J K L M N O is stored
// as its Transpose
// X[0] = A D G J M X[2]|
// X[1] = B E H K N <-------> | /X[1]
// X[2] = C F I L O axes |/
// high low 0------ X[0]
// Axes are stored conveniently as b-bit integers.
// Author: John Skilling 20 Apr 2001 to 11 Oct 2003