Kevin Ryde > Math-PlanePath-116 > Math::PlanePath::HilbertCurve

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NAME ^

Math::PlanePath::HilbertCurve -- 2x2 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,

           ...
        |   |
      7 |  63--62  49--48--47  44--43--42
        |       |   |       |   |       |
      6 |  60--61  50--51  46--45  40--41
        |   |           |           |
      5 |  59  56--55  52  33--34  39--38
        |   |   |   |   |   |   |       |
      4 |  58--57  54--53  32  35--36--37
        |                   |
      3 |   5---6   9--10  31  28--27--26
        |   |   |   |   |   |   |       |
      2 |   4   7---8  11  30--29  24--25
        |   |           |           |
      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 N=0 to N=3, then four of those are put together in an upside-down U as

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

The orientation of the sub parts ensure the starts and ends are adjacent, so 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.

See examples/hilbert-path.pl in the Math-PlanePath sources for a sample program printing the path pattern in ascii.

Level Ranges

Within a power-of-2 square 2x2, 4x4, 8x8, 16x16 etc (2^k)x(2^k) at the origin, 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 from xy_to_n()).

On the X=Y diagonal N=0,2,8,10,32,etc is the integers using only digits 0 and 2 in base 4, or equivalently have even-numbered bits 0, like x0y0...z0.

Locality

The Hilbert curve is fairly well localized in the sense that 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 curve N as an index. A search through an 2-D region is then usually a fairly modest linear search through N values. rect_to_n_range() gives exact N range for a rectangle, or see "Rectangle to N Range" below for calculating on any shape.

The N range can be large when crossing 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 X,Y binary fractions. The code here doesn't do that, but could be pressed into service by moving the binary point in X and Y an even number of places, the same in each. (An odd number of bits would require swapping X,Y to compensate for the alternating transpose in part 0.) The resulting integer N is then divided down by a corresponding multiple-of-4 binary places.

FUNCTIONS ^

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

$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)

The returned range is exact, meaning $n_lo and $n_hi are the smallest and biggest 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. Then 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 that sub-part. William 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 works because there's no "reverse" sections, or since the curve is the same forward and reverse. Low to high saves locating the top bits of N, but if using bignums then the bitwise inverts of the full X,Y values will be much more work.

X,Y to N

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).

Rectangle to N Range

An easy over-estimate of the maximum N in a region can be had by finding 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, so

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

Or equivalently rounding down to the next lower power of 2,

    find highest k with 2^k <= max(X,Y)
    N_max = 2^(2*(k+1)) - 1

An exact N range can be found by following the high to low N to X,Y procedure above. Start at the 2^(2k) bit pair position in an 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 for N which overlaps.

The biggest and smallest N digit for a sub-part can be found with a lookup table. The X range might cover one or both sub-parts, and the Y range similarly, for a total 9 possible configurations. Then a table of state+coverage -> digit gives the minimum and maximum N bit-pair, and state+digit gives a new state the same as X,Y to N.

Biggest and smallest N must be calculated with separate state and X,Y values since they track down different N bits and thus different states. But they take the same number of steps from an enclosing level down to level 0 and can thus be done in a single loop.

The N range for any shape can be found this way, not just a rectangle like rect_to_n_range(). At each level the procedure only depends on asking which combination of the four sub-parts overlaps some of the target area.

Direction

Each step between successive N values is always 1 up, down, left or right. The next direction can be calculated from N in the high-to-low procedure above by watching for the lowest non-3 digit and noting the direction from that digit towards digit+1. That can be had from the state+digit -> X,Y table looking up digit and digit+1, or alternatively a further table encoding state+digit -> direction.

The reason for taking only the lowest non-3 digit is that in a 3 sub-part the direction it goes is determined by the next higher level. For example at N=11 the direction is down for the inverted-U of the next higher level N=0,4,8,12.

This non-3 (or non whatever highest digit) is a general procedure and can be used on any state-based high-to-low procedure of self-similar curves. In the current code it's used to apply a fractional part of N in the correct direction but is not otherwise made directly available.

Because the Hilbert curve has no "reversal" sections it also works to build a direction from low to high N digits. 1 and 2 digits make no change to the orientation, 0 digit is a transpose, and a 3 digit is a rotate and transpose, except that low 3s are transpose-only (no rotate) for the same reason as taking the lowest non-3 above.

Jorg Arndt in the fxtbook above notes the direction can be obtained just by counting 3s in n and -n (the twos-complement). The only thing to note is that the numbering there starts n=1, unlike the PlanePath starting N=0, so it becomes

    N+1 count 3s  / 0 mod 2   S or E
                  \ 1 mod 2   N or W

    -(N+1) count 3s  / 0 mod 2   N or E
                     \ 1 mod 2   S or W

For the twos-complement negation an even number of base-4 digits of N must be taken. Because -(N+1) = ~N, ie. a ones-complement, the second part is also

    N count 0s          / 0 mod 2   N or E
    in even num digits  \ 1 mod 2   S or W

Putting the two together then

    N count 0s   N+1 count 3s    direction (0=E,1=N,etc)
    in base 4    in base 4

      0 mod 2      0 mod 2          0
      1 mod 2      0 mod 2          3
      0 mod 2      1 mod 2          1
      1 mod 2      1 mod 2          2

Segments in Direction

The number of segments in each direction is calculated in

Sergey Kitaev, Toufik Mansour and Patrice Séébold, "Generating the Peano Curve and Counting Occurrences of Some Patterns", Journal of Automata, Languages and Combinatorics, volume 9, number 4, 2004, pages 439-455. https://personal.cis.strath.ac.uk/sergey.kitaev/publications.html https://personal.cis.strath.ac.uk/sergey.kitaev/index_files/Papers/peano.ps

(Preprint as Sergey Kitaev and Toufik Mansour, "The Peano Curve and Counting Occurrences of Some Patterns", October 2002. http://arxiv.org/abs/math/0210268/, version 1.)

Their form is based on keeping the top-most U shape fixed and expanding sub-parts. This means the end segments alternate vertical and horizontal in successive expansion levels.

    direction            k=1              2       2
      1 to 4                            *---*   *---*
                           2           1|  3|   |1  |3
        1                *---*          *   *---*   *
        |               1|   |3        1| 4   2   4 |3
    4--- ---2            *   *          *---*   *---*
        |                                  1|   |3       k=2
        3                               *---*   *---*
                                          2       2

    count segments in direction, for k >= 1
    d(1,k) = 4^(k-1)                = 1,4,16,64,256,1024,4096,...
    d(2,k) = 4^(k-1) + 2^(k-1) - 1  = 1,5,19,71,271,1055,4159,...
    d(3,k) = 4^(k-1)                = 1,4,16,64,256,1024,4096,...
    d(4,k) = 4^(k-1) - 2^(k-1)      = 0,2,12,56,240, 992,4032,...

    total segments d(1,k)+d(2,k)+d(3,k)+d(4,k) = 4^k - 1

The form in the path here keeps the first segment direction fixed. This means a transpose 1<->2 and 3<->4 in odd levels. The result is to take the alternate d values as follows. For k=0 there is a single point N=0 so no line segments at all and so c(dir,0)=0.

    points N=0 to N=4^k-1 inclusive

    c(1,k) = / 0                        if k=0
     North   | 4^(k-1) + 2^(k-1) - 1    if k odd >= 1
             \ 4^(k-1)                  if k even >= 2
      = 0, 1, 4, 19, 64, 271, 1024, 4159, 16384, ...


    c(2,k) = / 0                        if k=0
     East    | 4^(k-1)                  if k odd >= 1
             \ 4^(k-1) + 2^(k-1) - 1    if k even >= 2
      = 0, 1, 5, 16, 71, 256, 1055, 4096, 16511, ...

    c(3,k) = / 0                        if k=0
     South   | 4^(k-1) - 2^(k-1)        if k odd >= 1
             \ 4^(k-1)                  if k even >= 2
      = 0, 0, 4, 12, 64, 240, 1024, 4032, 16384, ...

    c(4,k) = / 0                        if k=0
     West    | 4^(k-1)                  if k odd >= 1
             \ 4^(k-1) - 2^(k-1)        if k even >= 2
      = 0, 1, 2, 16, 56, 256, 992, 4096, 16256, ...

The segment N=4^k-1 to N=4^k is North (direction 1) when k odd, or East (direction 2) when k even. That could be added to the respective cases in c(1,k) and c(2,k) if desired.

Hamming Distance

The Hamming distance between integers X and Y is the number of bit positions where the two values differ when written in binary. On the Hilbert curve each bit-pair of N becomes a bit of X and a bit of Y,

       N      X   Y
    ------   --- ---
    0 = 00    0   0
    1 = 01    1   0     <- difference 1 bit
    2 = 10    1   1
    3 = 11    0   1     <- difference 1 bit

So the Hamming distance for N=0to3 is 1 at N=1 and N=3. As higher levels these the X,Y bits may be transposed (swapped) or rotated by 180 or both. A transpose swapping X<->Y doesn't change the bit difference. A rotate by 180 is a flip 0<->1 of the bit in each X and Y, so that doesn't change the bit difference either.

On that basis the Hamming distance X,Y is the number of base4 digits of N which are 01 or 11. If bit positions are counted from 0 for the least significant bit then

    HammingDist(X,Y) = count 1-bits at even bit positions in N    

See also "Hamming Distance" in Math::PlanePath::CornerReplicate which has the same formula, but arising directly from 01 or 11, no transpose or rotate.

OEIS ^

This path is in Sloane's OEIS in many forms,

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

    A059252    Y coord
    A059253    X coord
    A059261    X+Y
    A059285    X-Y
    A163547    X^2+Y^2 radius squared
    A139351    HammingDist(X,Y), count 1-bits at even positions in N

    A163365    sum N on diagonal
    A163477    sum N on diagonal, divided by 4
    A163482    N values on X axis
    A163483    N values on Y axis
    A062880    N values on diagonal X=Y (digits 0,2 in base 4)

    A163538    dX -1,0,1 change in X
    A163539    dY -1,0,1 change in Y
    A163540    absolute direction of each step (0=E,1=S,2=W,3=N)
    A163541    absolute direction, swapped X,Y
    A163542    relative direction (ahead=0,right=1,left=2)
    A163543    relative direction, swapped X,Y

    A083885    count East segments N=0 to N=4^k

    A163900    distance dX^2+dY^2 between Hilbert and ZOrder
    A165464    distance dX^2+dY^2 between Hilbert and Peano
    A165466    distance dX^2+dY^2 between Hilbert and transposed Peano
    A165465    N where Hilbert and Peano have same X,Y
    A165467    N where Hilbert and Peano have transposed same X,Y

The following take points of the plane in various orders, each value in the sequence being the N of the Hilbert curve at those positions.

    A163355    N by the ZOrderCurve points sequence
    A163356      inverse, ZOrderCurve by Hilbert points order
    A166041    N by the PeanoCurve points sequence
    A166042      inverse, PeanoCurve N by Hilbert points order
    A163357    N by diagonals like Math::PlanePath::Diagonals with
               first Hilbert step along same axis the diagonals start
    A163358      inverse
    A163359    N by diagonals, transposed start along the opposite axis
    A163360      inverse
    A163361    A163357 + 1, numbering the Hilbert N's from N=1
    A163362      inverse
    A163363    A163355 + 1, numbering the Hilbert N's from N=1
    A163364     inverse

These sequences are permutations of the integers since all X,Y positions of the first quadrant are covered by each path (Hilbert, ZOrder, Peano). The inverse permutations can be thought of taking X,Y positions in the Hilbert order and asking what N the ZOrder, Peano or Diagonals path would put there.

The A163355 permutation by ZOrderCurve can be considered for repeats or cycles,

    A163905    ZOrderCurve permutation A163355 applied twice
    A163915    ZOrderCurve permutation A163355 applied three times
    A163901    fixed points (N where X,Y same in both curves)
    A163902    2-cycle points
    A163903    3-cycle points
    A163890    cycle lengths, points by N
    A163904    cycle lengths, points by diagonals
    A163910    count of cycles in 4^k blocks
    A163911    max cycle length in 4^k blocks
    A163912    LCM of cycle lengths in 4^k blocks
    A163914    count of 3-cycles in 4^k blocks
    A163909      those counts for even k only
    A163891    N of previously unseen cycle length
    A163893      first differences of those A163891
    A163894    smallest value not an n-cycle
    A163895      position of new high in A163894
    A163896      value of new high in A163894

    A163907    ZOrderCurve permutation twice, on points by diagonals
    A163908      inverse of this

See examples/hilbert-oeis.pl in the Math-PlanePath sources for a sample program printing the A163359 permutation values.

SEE ALSO ^

Math::PlanePath, Math::PlanePath::PeanoCurve, Math::PlanePath::ZOrderCurve, Math::PlanePath::BetaOmega, Math::PlanePath::KochCurve

Math::Curve::Hilbert, Algorithm::SpatialIndex::Strategy::QuadTree

David Hilbert, "Ueber die stetige Abbildung einer Line auf ein Flächenstück", Mathematische Annalen, volume 38, number 3, p459-460, DOI 10.1007/BF01199431. http://www.springerlink.com/content/v1u6427kk33k8j56/ http://notendur.hi.is/oddur/hilbert/gcs-wrapper-1.pdf

HOME PAGE ^

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

LICENSE ^

Copyright 2010, 2011, 2012, 2013, 2014 Kevin Ryde

This file is part of Math-PlanePath.

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/>.

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