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

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/>.