Provided by: libmath-planepath-perl_122-1_all 
NAME
Math::PlanePath::QuadricCurve -- eight segment zig-zag
SYNOPSIS
use Math::PlanePath::QuadricCurve;
my $path = Math::PlanePath::QuadricCurve->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This is a self-similar zig-zag of eight segments,
18-19 5
| |
16-17 20 23-24 4
| | | |
15-14 21-22 25-26 3
| |
11-12-13 29-28-27 2
| |
2--3 10--9 30-31 58-59 ... 1
| | | | | | |
0--1 4 7--8 32 56-57 60 63-64 <- Y=0
| | | | | |
5--6 33-34 55-54 61-62 -1
| |
37-36-35 51-52-53 -2
| |
38-39 42-43 50-49 -3
| | | |
40-41 44 47-48 -4
| |
45-46 -5
^
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
The base figure is the initial N=0 to N=8,
2---3
| |
0---1 4 7---8
| |
5---6
It then repeats, turned to follow edge directions, so N=8 to N=16 is the same shape going
upwards, then N=16 to N=24 across, N=24 to N=32 downwards, etc.
The result is the base at ever greater scale extending to the right and with wiggly lines
making up the segments. The wiggles don't overlap.
The name "QuadricCurve" here is a slight mistake. Mandelbrot ("Fractal Geometry of
Nature" 1982 page 50) calls any islands initiated from a square "quadric", only one of
which is with sides by this eight segment expansion. This curve expansion also appears
(unnamed) in Mandelbrot's "How Long is the Coast of Britain", 1967.
Level Ranges
A given replication extends to
Nlevel = 8^level
X = 4^level
Y = 0
Ymax = 4^0 + 4^1 + ... + 4^level # 11...11 in base 4
= (4^(level+1) - 1) / 3
Ymin = - Ymax
Turn
The sequence of turns made by the curve is straightforward. In the base 8 (octal)
representation of N, the lowest non-zero digit gives the turn
low digit turn (degrees)
--------- --------------
1 +90
2 -90
3 -90
4 0
5 +90
6 +90
7 -90
When the least significant digit is non-zero it determines the turn, to make the base N=0
to N=8 shape. When the low digit is zero it's instead the next level up, the
N=0,8,16,24,etc shape which is in control, applying a turn for the subsequent base part.
So for example at N=16 = 20 octal 20 is a turn -90 degrees.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$path = Math::PlanePath::QuadricCurve->new ()"
Create and return a new path object.
"($x,$y) = $path->n_to_xy ($n)"
Return the X,Y coordinates of point number $n on the path. Points begin at 0 and if
"$n < 0" then the return is an empty list.
Level Methods
"($n_lo, $n_hi) = $path->level_to_n_range($level)"
Return "(0, 8**$level)".
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
<http://oeis.org/A133851> (etc)
A133851 Y at N=2^k, being successive powers 2^j at k=1mod4
SEE ALSO
Math::PlanePath, Math::PlanePath::QuadricIslands, Math::PlanePath::KochCurve
Math::Fractal::Curve -- its examples/generator4.pl is this curve
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015 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/>.