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Math::PlanePath::QuadricIslands(3pm) User Contributed Perl Documentation Math::PlanePath::QuadricIslands(3pm)

Math::PlanePath::QuadricIslands -- quadric curve rings

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

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.

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

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

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

"($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 )

Math::PlanePath, Math::PlanePath::QuadricCurve, Math::PlanePath::KochSnowflakes, Math::PlanePath::GosperIslands

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

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/>.

2021-01-23 perl v5.32.0