NAME

       Math::PlanePath::QuadricIslands -- 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 )

SEE ALSO

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

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/>.