Provided by: libmath-planepath-perl_113-1_all 
NAME
Math::PlanePath::QuadricCurve -- eight segment zig-zag
SYNOPSIS
use Math::PlanePath::QuadricCurve;
my $path = Math::PlanePath::QuadricCurve->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This is a self-similar zig-zag of eight segments,
18-19 5
| |
16-17 20 23-24 4
| | | |
15-14 21-22 25-26 3
| |
11-12-13 29-28-27 2
| |
2--3 10--9 30-31 58-59 ... 1
| | | | | | |
0--1 4 7--8 32 56-57 60 63-64 <- Y=0
| | | | | |
5--6 33-34 55-54 61-62 -1
| |
37-36-35 51-52-53 -2
| |
38-39 42-43 50-49 -3
| | | |
40-41 44 47-48 -4
| |
45-46 -5
^
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
The base figure is the initial N=0 to N=8,
2---3
| |
0---1 4 7---8
| |
5---6
It then repeats, turned to follow edge directions, so N=8 to N=16 is the same shape going
upwards, then N=16 to N=24 across, N=24 to N=32 downwards, etc.
The result is the base at ever greater scale extending to the right and with wiggly lines
making up the segments. The wiggles don't overlap.
Level Ranges
A given replication extends to
Nlevel = 8^level
X = 4^level
Y = 0
Ymax = 4^0 + 4^1 + ... + 4^level # 11...11 in base 4
= (4^(level+1) - 1) / 3
Ymin = - Ymax
Turn
The sequence of turns made by the curve is straightforward. In the base 8 (octal)
representation of N, the lowest non-zero digit gives the turn
low digit turn (degrees)
--------- --------------
1 +90
2 -90
3 -90
4 0
5 +90
6 +90
7 -90
When the least significant digit is non-zero it determines the turn, to make the base N=0
to N=8 shape. When the low digit is zero it's instead the next level up, the
N=0,8,16,24,etc shape which is in control, applying a turn for the subsequent base part.
So for example at N=16 = 20 octal 20 is a turn -90 degrees.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$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.
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 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/>.