NAME

Math::PlanePath::HilbertSpiral -- 2x2 self-similar spiral

SYNOPSIS

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

DESCRIPTION

This is a Hilbert curve variation which fills the plane by spiralling around into negative X,Y on every second replication level.

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

The curve starts with the same N=0 to N=3 as the "HilbertCurve", then the following 2x2 blocks N=4 to N=15 go around in negative X,Y. The top-left corner for this negative direction is at Ntopleft=4^level-1 for an odd numbered level.

The parts of the curve in the X,Y negative parts are the same as the plain "HilbertCurve", just mirrored along the anti-diagonal. For example. N=4 to N=15

    HilbertSpiral             HilbertCurve
                  \        5---6   9--10
                   \       |   |   |   |
                    \      4   7---8  11
                     \                 |
      5-- 4           \           13--12
      |                \           |
      6-- 7             \         14--15
          |              \
      9-- 8  13--14       \
      |       |   |        \
     10--11--12  15

This mirroring has the effect of mapping

    HilbertCurve X,Y  ->  -Y,-X for HilbertSpiral

Notice the coordinate difference (-Y)-(-X) = X-Y so that difference, representing a projection onto the X=-Y opposite diagonal, is the same in both paths.

Level Ranges

Reckoning the initial N=0 to N=3 as level 1, a replication level extends to

    Nstart = 0
    Nlevel = 4^level - 1    (inclusive)
    Xmin = Ymin = - (4^floor(level/2) - 1) * 2 / 3
                = binary 1010...10
    Xmax = Ymax = (4^ceil(level/2) - 1) / 3
                = binary 10101...01
    width = height = Xmax - Xmin
                   = Ymax - Ymin
                   = 2^level - 1

The X,Y range doubles alternately above and below, so the result is a 1 bit going alternately to the max or min, starting with the max for level 1.

    level     X,Ymin   binary      X,Ymax  binary
    -----     ---------------      --------------
      0         0                    0
      1         0          0         1 =       1
      2        -2 =      -10         1 =      01
      3        -2 =     -010         5 =     101
      4       -10 =    -1010         5 =    0101
      5       -10 =   -01010        21 =   10101
      6       -42 =  -101010        21 =  010101
      7       -42 = -0101010        85 = 1010101

The power-of-4 formulas above for Ymin/Ymax have the effect of producing alternating bit patterns like this.

This is the same sort of level range as "BetaOmega" has on its Y coordinate, but on this "HilbertSpiral" it applies to both X and Y.

FUNCTIONS

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

"$path = Math::PlanePath::HilbertSpiral->new ()": Create and return a new path object.
"($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.
"($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.

Level Methods

"($n_lo, $n_hi) = $path->level_to_n_range($level)": Return "(0, 4**$level - 1)".

OEIS

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

<http://oeis.org/A059285> (etc)

    A059285    X-Y coordinate diff

The difference X-Y is the same as the "HilbertCurve", since the "negative" spiral parts are mirrored across the X=-Y anti-diagonal, which means coordinates (-Y,-X) and -Y-(-X) = X-Y.

HOME PAGE

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

LICENSE

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/>.

2021-01-23

perl v5.32.0