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.
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
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 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/>.
