NAME

Math::PlanePath::HilbertCurve -- self-similar quadrant traversal

SYNOPSIS

 use Math::PlanePath::HilbertCurve;
 my $path = Math::PlanePath::HilbertCurve->new;
 my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

This path is an integer version of the curve described by David Hilbert in 1891 for filling a unit square. It traverses a quadrant of the plane one step at a time in a self-similar 2x2 pattern,

             ...
              |
      y=7    63--62  49--48--47  44--43--42
                  |   |       |   |       |
      y=6    60--61  50--51  46--45  40--41
              |           |           |
      y=5    59  56--55  52  33--34  39--38
              |   |   |   |   |   |       |
      y=4    58--57  54--53  32  35--36--37
                              |
      y=3     5---6   9--10  31  28--27--26
              |   |   |   |   |   |       |
      y=2     4   7---8  11  30--29  24--25
              |           |           |
      y=1     3---2  13--12  17--18  23--22
                  |   |       |   |       |
      y=0     0---1  14--15--16  19--20--21

            x=0   1   2   3   4   5   6   7

The start is a sideways U shape per N=0 to N=3, and then four of those are put together in an upside-down U as follows,

    5,6    9,10
    4,7--- 8,11
      |      |
    3,2   13,12
    0,1   14,15--

The orientation of the sub parts are chosen so the starts and ends are adjacent, 3 next to 4, 7 next to 8, and 11 next to 12.

The process repeats, doubling in size each time and alternately sideways or upside-down U with invert and/or transpose as necessary in the sub-parts.

The pattern is sometimes drawn with the first step 0->1 upwards instead of to the right. Right is used here since that's what most of the other PlanePaths do. Swap X and Y for upwards first instead.

Within a power-of-2 square 2x2, 4x4, 8x8, 16x16 etc (2^k)x(2^k), all the N values 0 to 2^(2*k)-1 are within the square. The maximum 3, 15, 63, 255 etc 2^(2*k)-1 is alternately at the top left or bottom right corner.

Because each step is by 1, the distance along the curve between two X,Y points is the difference in their N values (as given by xy_to_n).

Locality

The Hilbert curve is fairly well localized, so a small rectangle (or other shape) is usually a small range of N. This property is used in some database systems to store X,Y coordinates with the Hilbert N as an index. A search through an X,Y region is then usually a fairly modest linear N search. rect_to_n_range gives N bounds for a rectangle, or see "N Range" below for calculating on any shape.

The N range can be large when crossing Hilbert sub-parts. In the sample above it can be seen for instance adjacent points X=0,Y=3 and X=0,Y=4 have rather widely spaced N values 5 and 58.

Fractional X,Y values can be indexed by extending the N calculation down into the X,Y binary fractions. The code here doesn't do this, but can be pressed into service by moving the binary point in X,Y down even number of places, the same amount in each, and in the resulting integer N shifting back up by a corresponding multiple of 4 binary places.

FUNCTIONS

$path = Math::PlanePath::HilbertCurve->new ()

Create and return a new path object.

($x,$y) = $path->n_to_xy ($n)

Return the X,Y coordinates of point number $n on the path. Points begin at 0 and if $n < 0 then the return is an empty list.

Fractional positions give an X,Y position along a straight line between the integer positions. Integer positions are always just 1 apart either horizontally or vertically, so the effect is that the fraction part is an offset along either X or Y.

$n = $path->xy_to_n ($x,$y)

Return an integer point number for coordinates $x,$y. Each integer N is considered the centre of a unit square and an $x,$y within that square returns N.

($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)

Return a range of N values which occur in a rectangle with corners at $x1,$y1 and $x2,$y2. If the X,Y values are not integers then the curve is treated as unit squares centred on each integer point and squares which are partly covered by the given rectangle are included.

The returned range is exact, meaning $n_lo is the smallest in the rectangle and $n_hi is the biggest. Of course not all the N's in that range are necessarily in the rectangle.

FORMULAS

N to X,Y

Converting N to X,Y coordinates is reasonably straightforward. The top two bits of N is a configuration

    3--2                    1--2
       |    or transpose    |  |
    0--1                    0  3

according to whether it's an odd or even bit-pair position. Within each of the "3" sub-parts there's also inverted forms

    1--0        3  0
    |           |  |
    2--3        2--1

Working N from high to low with a state variable can record whether there's a transpose, an invert, or both, being four states altogether. A bit pair 0,1,2,3 from N then gives a bit each of X,Y according to the configuration, and a new state which is the orientation of the sub-part. Gosper's HAKMEM item 115 has this with tables for the state and X,Y bits,

    http://www.inwap.com/pdp10/hbaker/hakmem/topology.html#item115

And C++ code based on that in Jorg Arndt's book,

    http://www.jjj.de/fxt/#fxtbook (section 1.31.1)

It also works to process N from low to high, at each stage applying any transpose (swap X,Y) and/or invert (bitwise NOT) to the low X,Y bits generated so far. This saves locating the top bits of N, but if using bignums then bitwise inverts will be much more work.

X,Y to N

The reverse X,Y to N can follow the table approach from high to low taking one bit from X and Y each time. The state table of N-pair -> X-bit,Y-bit is reversible, and a new state is based on the N-pair thus obtained (or could be based on the X,Y bits if that mapping is combined into the state transition table).

The current code is a mixture of the low to high for n_to_xy but the table high to low for the reverse xy_to_n.

N Range

An easy over-estimate of the maximum N in a region can be had by going to the next bigger (2^k)x(2^k) square enclosing the region. This means the biggest X or Y rounded up to the next power of 2 (perhaps by a "count leading zeros" CPU instruction), so

    find k with 2^k > max(X,Y)
    N_max = 2^(2k) - 1

An exact N range can be found by following the high to low N to X,Y procedure. Start at the 2^(2k) bit pair position in N bigger than the desired region and choose 2 bits for N to give a bit each of X and Y. The X,Y bits are based on the state table as above and the bits chosen for N are those for which the resulting X,Y sub-square overlaps some of the target region. The smallest N similarly, choosing the smallest bit pair.

Biggest and smallest N must be calculated separately as they track down different N bits and thus different state transitions. The N range for any shape can be done this way, not just a rectangle like rect_to_n_range, since it only depends on asking when a sub-square overlaps the target area.

Direction

Each step between successive N values is by 1 up, down, left or right. The next direction can be calculated from the N position with on some base-4 digit-3s parity of N and -N (twos complement). C++ code in Jorg Arndt's fxtbook per above.

OEIS

This Hilbert Curve path is in Sloane's OEIS in several forms,

    http://oeis.org/A059252  (etc)

    A059252    Y coord    \ if first move horizontal
    A059253    X coord    / per the code here
    A163365    diagonal sums
    A163477    diagonal sums, divided by 4
    A163482    row at Y=0
    A163483    column at X=0
    A163538    X change -1,0,1
    A163539    Y change -1,0,1
    A163540    absolute direction of each step (up,down,left,right)
    A163541    absolute direction, transpose X,Y
    A163542    relative direction (ahead,left,right)
    A163543    relative direction, transpose X,Y

And taking squares of the plane in various orders, each value the N of the Hilbert curve at those positions.

    A163355    in the ZOrderCurve sequence
    A163357    in diagonals like Math::PlanePath::Diagonals with
               first Hilbert step along same axis the diagonals start
    A163359    in diagonals, transposed start along the opposite axis
    A163361    A163357 + 1, numbering the Hilbert N's from N=1
    A163363    A163355 + 1, numbering the Hilbert N's from N=1

The sequences are in each case permutations of the integers since all X,Y positions are reached eventually. The inverses are

    A163356    inverse of A163355
    A163358    inverse of A163357
    A163360    inverse of A163359
    A163362    inverse of A163361

SEE ALSO

Math::PlanePath, Math::PlanePath::PeanoCurve, Math::PlanePath::ZOrderCurve

Math::Curve::Hilbert, Algorithm::QuadTree

David Hilbert, "Ueber die stetige Abbildung einer Line auf ein Flächenstück", Mathematische Annalen, volume 38, number 3, p459-460,

    http://www.springerlink.com/content/v1u6427kk33k8j56/
    DOI 10.1007/BF01199431
    http://notendur.hi.is/oddur/hilbert/gcs-wrapper-1.pdf

HOME PAGE

http://user42.tuxfamily.org/math-planepath/index.html

LICENSE

Math-PlanePath is Copyright 2010, 2011 Kevin Ryde

Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.

Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.

cpanm Math::PlanePath

perl -MCPAN -e shell
install Math::PlanePath