Provided by: libmath-planepath-perl_129-1_all 
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/>.
perl v5.32.0 2021-01-23 Math::PlanePath::QuadricIslands(3pm)