/ 
Math-PlanePath-129
River stage one • 3 direct dependents • 4 total dependents
/ Math::PlanePath::HilbertSpiral

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

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.

SEE ALSO

Math::PlanePath, Math::PlanePath::HilbertCurve, Math::PlanePath::BetaOmega

HOME PAGE

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

LICENSE

Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 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/>.