NAME
Math::PlanePath::R5DragonCurve -- radix 5 dragon curve
SYNOPSIS
use Math::PlanePath::R5DragonCurve;
my $path = Math::PlanePath::R5DragonCurve->new;
my ($x, $y) = $path->n_to_xy (123);DESCRIPTION
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 1The base figure is an "S" shape
4----5
|
3----2
|
0----1which 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----1The 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 1The curve never crosses itself. The vertices touch as corners like N=4 and N=8 above, but no edges repeat.
Spiralling
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^levelThat 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,-2 190.3Arms
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 => 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
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---18/34With 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.
Tiling
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.
FUNCTIONS
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
armsparameter 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
$non the path. Points begin at 0 and if$n < 0then the return is an empty list.Fractional
$ngives 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$x,$ythen returnundef.The curve can visit an
$x,$ytwice. 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.
FORMULAS
Turns
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 rightEssentially 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^kThe 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.
OEIS
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+1SEE ALSO
Math::PlanePath, Math::PlanePath::DragonCurve, Math::PlanePath::TerdragonCurve
HOME PAGE
http://user42.tuxfamily.org/math-planepath/index.html
LICENSE
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.
You should have received a copy of the GNU General Public License along with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.