Provided by: libmath-planepath-perl_113-1_all bug

NAME

       Math::PlanePath::ZOrderCurve -- alternate digits to X and Y

SYNOPSIS

        use Math::PlanePath::ZOrderCurve;

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

        # or another radix digits ...
        my $path3 = Math::PlanePath::ZOrderCurve->new (radix => 3);

DESCRIPTION

       This path puts points in a self-similar Z pattern described by G.M. Morton,

             7  |   42  43  46  47  58  59  62  63
             6  |   40  41  44  45  56  57  60  61
             5  |   34  35  38  39  50  51  54  55
             4  |   32  33  36  37  48  49  52  53
             3  |   10  11  14  15  26  27  30  31
             2  |    8   9  12  13  24  25  28  29
             1  |    2   3   6   7  18  19  22  23
            Y=0 |    0   1   4   5  16  17  20  21  64  ...
                +---------------------------------------
                   X=0   1   2   3   4   5   6   7   8

       The first four points make a "Z" shape if written with Y going downwards (inverted if
       drawn upwards as above),

            0---1       Y=0
               /
             /
            2---3       Y=1

       Then groups of those are arranged as a further Z, etc, doubling in size each time.

            0   1      4   5       Y=0
            2   3 ---  6   7       Y=1
                    /
                   /
                  /
            8   9 --- 12  13       Y=2
           10  11     14  15       Y=3

       Within an 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 top right corner 3, 15, 63, 255 etc of each is the
       2^(2*k)-1 maximum.

       Along the X axis N=0,1,4,5,16,17,etc is the integers with only digits 0,1 in base 4.
       Along the Y axis N=0,2,8,10,32,etc is the integers with only digits 0,2 in base 4.  And
       along the X=Y diagonal N=0,3,12,15,etc is digits 0,3 in base 4.

       In the base Z pattern it can be seen that transposing to Y,X means swapping parts 1 and 2.
       This applies in the sub-parts too so in general if N is at X,Y then changing base 4 digits
       1<->2 gives the N at the transpose Y,X.  For example N=22 at X=6,Y=1 is base-4 "112",
       change 1<->2 is "221" for N=41 at X=1,Y=6.

   Power of 2 Values
       Plotting N values related to powers of 2 can come out as interesting patterns.  For
       example displaying the N's which have no digit 3 in their base 4 representation gives

           *
           * *
           *   *
           * * * *
           *       *
           * *     * *
           *   *   *   *
           * * * * * * * *
           *               *
           * *             * *
           *   *           *   *
           * * * *         * * * *
           *       *       *       *
           * *     * *     * *     * *
           *   *   *   *   *   *   *   *
           * * * * * * * * * * * * * * * *

       The 0,1,2 and not 3 makes a little 2x2 "L" at the bottom left, then repeating at 4x4 with
       again the whole "3" position undrawn, and so on.  This is the Sierpinski triangle (a
       rotated version of Math::PlanePath::SierpinskiTriangle).  The blanks are also a visual
       representation of 1-in-4 cross-products saved by recursive use of the Karatsuba
       multiplication algorithm.

       Plotting the fibbinary numbers (eg. Math::NumSeq::Fibbinary) which are N values with no
       adjacent 1 bits in binary makes an attractive tree-like pattern,

           *
           **
           *
           ****
           *
           **
           *   *
           ********
           *
           **
           *
           ****
           *       *
           **      **
           *   *   *   *
           ****************
           *                               *
           **                              **
           *                               *
           ****                            ****
           *                               *
           **                              **
           *   *                           *   *
           ********                        ********
           *               *               *               *
           **              **              **              **
           *               *               *               *
           ****            ****            ****            ****
           *       *       *       *       *       *       *       *
           **      **      **      **      **      **      **      **
           *   *   *   *   *   *   *   *   *   *   *   *   *   *   *   *
           ****************************************************************

       The horizontals arise from N=...0a0b0c for bits a,b,c so Y=...000 and X=...abc, making
       those N values adjacent.  Similarly N=...a0b0c0 for a vertical.

   Radix
       The "radix" parameter can do the same N <-> X/Y digit splitting in a higher base.  For
       example radix 3 makes 3x3 groupings,

            radix => 3

             5  |  33  34  35  42  43  44
             4  |  30  31  32  39  40  41
             3  |  27  28  29  36  37  38  45  ...
             2  |   6   7   8  15  16  17  24  25  26
             1  |   3   4   5  12  13  14  21  22  23
            Y=0 |   0   1   2   9  10  11  18  19  20
                +--------------------------------------
                  X=0   1   2   3   4   5   6   7   8

       Along the X axis N=0,1,2,9,10,11,etc is integers with only digits 0,1,2 in base 9.  Along
       the Y axis digits 0,3,6, and along the X=Y diagonal digits 0,4,8.  In general for a given
       radix it's base R*R with the R many digits of the first RxR block.

FUNCTIONS

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

       "$path = Math::PlanePath::ZOrderCurve->new ()"
       "$path = Math::PlanePath::ZOrderCurve->new (radix => $r)"
           Create and return a new path object.  The optional "radix" parameter gives the base
           for digit splitting (the default is binary, radix 2).

       "($x,$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.  The lines don't overlap, but the lines between bit squares soon become
           rather long and probably of very limited use.

       "$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
       The coordinate calculation is simple.  The bits of X and Y are every second bit of N.  So
       if N = binary 101010 then X=000 and Y=111 in binary, which is the N=42 shown above at
       X=0,Y=7.

       With the "radix" parameter the digits are treated likewise, in the given radix rather than
       binary.

       If N includes a fraction part then it's applied to a straight line towards point N+1.  The
       +1 of N+1 changes X and Y according to how many low radix-1 digits there are in N, and
       thus in X and Y.  In general if the lowest non radix-1 is in X then

           dX=1
           dY = - (R^pos - 1)           # pos=0 for lowest digit

       The simplest case is when the lowest digit of N is not radix-1, so dX=1,dY=0 across.

       If the lowest non radix-1 is in Y then

           dX = - (R^(pos+1) - 1)       # pos=0 for lowest digit
           dY = 1

       If all digits of X and Y are radix-1 then the implicit 0 above the top of X is considered
       the lowest non radix-1 and so the first case applies.  In the radix=2 above this happens
       for instance at N=15 binary 1111 so X = binary 11 and Y = binary 11.  The 0 above the top
       of X is at pos=2 so dX=1, dY=-(2^2-1)=-3.

   Rectangle to N Range
       Within each row the N values increase as X increases, and within each column N increases
       with increasing Y (for all "radix" parameters).

       So for a given rectangle the smallest N is at the lower left corner (smallest X and
       smallest Y), and the biggest N is at the upper right (biggest X and biggest Y).

OEIS

       This path is in Sloane's Online Encyclopedia of Integer Sequences in various forms,

           <http://oeis.org/A059905> (etc)

           radix=2
             A059905    X coordinate
             A059906    Y coordinate

             A000695    N on X axis       (base 4 digits 0,1 only)
             A062880    N on Y axis       (base 4 digits 0,2 only)
             A001196    N on X=Y diagonal (base 4 digits 0,3 only)

             A057300    permutation N at transpose Y,X (swap bit pairs)

           radix=3
             A163325    X coordinate
             A163326    Y coordinate
             A037314    N on X axis (base 9 digits 0,1,2)
             A163327    permutation N at transpose Y,X (swap trit pairs)

           radix=4
             A126006    permutation N at transpose Y,X (swap digit pairs)

           radix=10
             A080463    X+Y of radix=10 (from N=1 onwards)
             A080464    X*Y of radix=10 (from N=10 onwards)
             A080465    abs(X-Y), from N=10 onwards
             A051022    N on X axis (base 100 digits 0 to 9)

           radix=16
             A217558    permutation N at transpose Y,X (swap digit pairs)

       And taking X,Y points in the Diagonals sequence then the value of the following sequences
       is the N of the "ZOrderCurve" at those positions.

           radix=2
             A054238    numbering by diagonals, from same axis as first step
             A054239      inverse permutation

           radix=3
             A163328    numbering by diagonals, same axis as first step
             A163329      inverse permutation
             A163330    numbering by diagonals, opp axis as first step
             A163331      inverse permutation

       "Math::PlanePath::Diagonals" numbers points from the Y axis down, which is the opposite
       axis to the "ZOrderCurve" first step along the X axis, so a transpose is needed to give
       A054238.

SEE ALSO

       Math::PlanePath, Math::PlanePath::PeanoCurve, Math::PlanePath::HilbertCurve,
       Math::PlanePath::ImaginaryBase, Math::PlanePath::CornerReplicate,
       Math::PlanePath::DigitGroups

       "http://www.jjj.de/fxt/#fxtbook" (section 1.31.2)

       Algorithm::QuadTree, DBIx::SpatialKeys

HOME PAGE

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

LICENSE

       Copyright 2010, 2011, 2012, 2013 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/>.