Math::PlanePath::R5DragonCurve -- radix 5 dragon curve
use Math::PlanePath::R5DragonCurve; my $path = Math::PlanePath::R5DragonCurve->new; my ($x, $y) = $path->n_to_xy (123);
This is the R5 dragon curve,
31-----30 27-----26 5 | | | | 32---29/33--28/24----25 4 | | 35---34/38--39/23----22 11-----10 7------6 3 | | | | | | 36---37/41--20/40--21/17--16/12---13/9----8/4-----5 2 | | | | | | --50 47---42/46--19/43----18 15-----14 3------2 1 | | | | | 49/53--48/64 45/65--44/68 69 0------1 <-Y=0 ^ ^ ^ ^ ^ ^ ^ ^ ^ -7 -6 -5 -4 -3 -2 -1 X=0 1
The base figure is an "S" shape
4----5 | 3----2 | 0----1
which then repeats in self-similar style, so N=5 to N=10 is a copy rotated +90 degrees, which is the angle of the N=1 to N=2 edge,
10 7----6 | | | <- repeat rotated +90 9---8,4---5 | 3----2 | 0----1
The shape of N=0,5,10,15,20,25 repeats the initial N=0 to N=5,
25 4 / / 10__ 3 / / ----___ 20__ / 5 2 ----__ / / 15 / 1 / 0 <-Y=0 ^ ^ ^ ^ ^ ^ -4 -3 -2 -1 X=0 1
The curve never crosses itself. The vertices touch as corners like N=4 and N=8 above, but no edges repeat.
The first step N=1 is to the right along the X axis and the path then slowly spirals anti-clockwise and progressively fatter. The end of each replication is
Nlevel = 5^level
That point is at atan(2,1)=63.43 degrees further around for each level,
Nlevel X,Y angle (degrees) ------ ----- ----- 1 1,0 0 5 2,1 63.4 25 -3,4 126.8 125 -11,-2 190.3
The curve fills a quarter of the plane and four copies mesh together perfectly rotated by 90, 180 and 270 degrees. The arms parameter can choose 1 to 4 such curve arms successively advancing.
arms
arms => 4 begins as follows. N=0,4,8,12,16,etc is the first arm (the same shape as the plain curve above), then N=1,5,9,13,17 the second, N=2,6,10,14 the third, etc.
arms => 4
arms => 4 16/32---20/63 | 21/60 9/56----5/12----8/59 | | | | 17/33--- 6/13--0/1/2/3---4/15---19/35 | | | | 10/57----7/14---11/58 23/62 | 22/61---18/34
With four arms every X,Y point is visited twice, except the origin 0,0 where all four begin. Every edge between the points is traversed once.
The little "S" shapes of the N=0to5 base shape tile the plane in the following pattern,
| | | | | | | | | +--+--+--+--+ +--+--+--+--+ +- | | | | | | | | | -+--+ +--+--+--+--+ +--+--+--+- | | | | | | | | -+--+--+--+ +--+--+--+--+ +--+- | | | | | | | | | -+ +--+--+--+--+ +--+--+--+--+ | | | | | | | | | -+--+--+ +--+--o--+--+ +--+--+- | | | | | | | | | +--+--+--+--+ +--+--+--+--+ +- | | | | | | | | | -+--+ +--+--+--+--+ +--+--+--+- | | | | | | | | -+--+--+--+ +--+--+--+--+ +--+- | | | | | | | | | -+ +--+--+--+--+ +--+--+--+--+ | | | | | | | | |
This is simply edge N=2mod5 to N=3mod5 omitted from each mod5 block. In each 2x1 block the "S" traverses 4 of the 6 edges and the way the curve meshes together traverses the other 2 edges in another brick, possibly a brick on another arm of the curve.
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::R5DragonCurve->new ()
$path = Math::PlanePath::R5DragonCurve->new (arms => 4)
Create and return a new path object.
The optional arms parameter can make 1 to 4 copies of the curve, each arm successively advancing.
($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.
$n
$n < 0
Fractional $n gives an X,Y position along a straight line between the integer positions.
$n = $path->xy_to_n ($x,$y)
Return the point number for coordinates $x,$y. If there's nothing at $x,$y then return undef.
$x,$y
undef
The curve can visit an $x,$y twice. In the current code the smallest of the these N values is returned. Is that the best way?
@n_list = $path->xy_to_n_list ($x,$y)
Return a list of N point numbers for coordinates $x,$y. There can be none, one or two N's for a given $x,$y.
$n = $path->n_start()
Return 0, the first N in the path.
At each point N the curve always turns 90 degrees either to the left or right, it never goes straight ahead. If N is written in base 5 then the lowest non-zero digit gives the turn
Ndigit Turn ------ ---- 1 left 2 left 3 right 4 right
Essentially at a point N=digit*5^level for digit=1,2,3,4 the turn follows the shape at that digit.
4*5^k----5^(k+1) | | 2*5^k----2*5^k | | 0------1*5^k
The first and last unit steps in each level are in the same direction, so at those endpoints it's the next level up which the turn.
The R5 dragon is in Sloane's Online Encyclopedia of Integer Sequences as,
http://oeis.org/A175337 A175337 -- turn 0=left,1=right by 90 degrees at N=n
Math::PlanePath, Math::PlanePath::DragonCurve, Math::PlanePath::TerdragonCurve
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2012 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.
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To install Math::PlanePath, copy and paste the appropriate command in to your terminal.
cpanm
cpanm Math::PlanePath
CPAN shell
perl -MCPAN -e shell install Math::PlanePath
For more information on module installation, please visit the detailed CPAN module installation guide.