Math::PlanePath::QuadricIslands(3) quadric curve rings

SYNOPSIS


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

DESCRIPTION

This path is concentric islands made from four sides each an eight segment zig-zag (per the "QuadicCurve" path). 

            27--26                     3
             |   |
        29--28  25  22--21             2
         |       |   |   |
        30--31  24--23  20--19         1
             | 4--3          |
    34--33--32    | 16--17--18     <- Y=0
     |         1--2  |
    35--36   7---8  15--14            -1
             |   |       |
         5---6   9  12--13            -2
                 |   |
                10--11                -3
                 ^
    -3  -2  -1  X=0  1   2   3   4

The initial figure is the square N=1,2,3,4 then for the next level each straight side expands to 4x longer and a zigzag like N=5 through N=13 and the further sides to N=36. The individual sides are levels of the "QuadricCurve" path. 

                                *---*
                                |   |
      *---*     becomes     *---*   *   *---*
                                    |   |
                                    *---*
         * <------ *
         |         ^
         |         |
         |         |
         v         |
         * ------> *

The name "QuadricIslands" here is a slight mistake. Mandelbrot (``Fractal Geometry of Nature'' 1982 page 50) calls any islands initiated from a square ``quadric'', not just this eight segment expansion. This curve also appears (unnamed) in Mandelbrot's ``How Long is the Coast of Britain'', 1967.

Level Ranges

Counting the innermost square as level 0, each ring is 

    length = 4 * 8^level     many points
    Nlevel = 1 + length[0] + ... + length[level-1]
           = (4*8^level + 3)/7
    Xstart = - 4^level / 2
    Ystart = - 4^level / 2

For example the lower partial ring shown above is level 2 starting N=(4*8^2+3)/7=37 at X=-(4^2)/2=-8,Y=-8.

The innermost square N=1,2,3,4 is on 0.5 coordinates, for example N=1 at X=-0.5,Y=-0.5. This is centred on the origin and consistent with the (4^level)/2. Points from N=5 onwards are integer X,Y. 

       4-------3    Y=+1/2
       |       |
       |   o   |
               |
       1-------2    Y=-1/2
    X=-1/2   X=+1/2

FUNCTIONS

See ``FUNCTIONS'' in Math::PlanePath for behaviour common to all path classes.
"$path = Math::PlanePath::QuadricIslands->new ()"
Create and return a new path object.

Level Methods

"($n_lo, $n_hi) = $path->level_to_n_range($level)"
Return per ``Level Ranges'' above, 

    ( ( 4 * 8**$level + 3) / 7,
      (32 * 8**$level - 4) / 7 )

HOME PAGE

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

LICENSE

Copyright 2011, 2012, 2013, 2014, 2015, 2016 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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