Math::PlanePath::DragonRounded -- dragon curve, with rounded corners
use Math::PlanePath::DragonRounded; my $path = Math::PlanePath::DragonRounded->new; my ($x, $y) = $path->n_to_xy (123);
This is a version of the dragon curve by Harter, Heighway, et al, done with two points per edge and skipping vertices so as to make rounded-off corners,
17-16 9--8 6 / \ / \ 18 15 10 7 5 | | | | 19 14 11 6 4 \ \ / \ 20-21 13-12 5--4 3 \ \ 22 3 2 | | 23 2 1 / / 33-32 25-24 . 0--1 Y=0 / \ / 34 31 26 -1 | | | 35 30 27 -2 \ \ / 36-37 29-28 44-45 -3 \ / \ 38 43 46 -4 | | | 39 42 47 -5 \ / / 40-41 49-48 -6 / 50 -7 | ... ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ -15-14-13-12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 ...
The two points on an edge have one of X or Y a multiple of 3 and the other Y or X at 1 mod 3 or 2 mod 3. For example N=19 and N=20 are on the X=-9 edge (a multiple of 3), and at Y=4 and Y=5 (1 and 2 mod 3).
The "rounding" of the corners ensures that for example N=13 and N=21 don't touch as they approach X=-6,Y=3. The curve always approaches vertices like this and never crosses itself.
The dragon curve fills a quarter of the plane and four copies mesh together rotated by 90, 180 and 270 degrees. The arms
parameter can choose 1 to 4 curve arms, successively advancing. For example arms => 4
gives
36-32 59-... 6 / \ / ... 40 28 55 5 | | | | 56 44 24 51 4 \ / \ \ 52-48 13--9 20-16 47-43 3 / \ \ \ 17 5 12 39 2 | | | | 21 1 8 35 1 / / / 29-25 6--2 0--4 27-31 <- Y=0 / / / 33 10 3 23 -1 | | | | 37 14 7 19 -2 \ \ \ / 41-45 18-22 11-15 50-54 -3 \ \ / \ 49 26 46 58 -4 | | | | 53 30 42 ... -5 / \ / ...-57 34-38 -6 ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6
With 4 arms like this all 3x3 blocks are visited, using 4 out of 9 points in each.
The points of this rounded curve correspond to the DragonMidpoint
with a little squish to turn each 6x6 block into a 4x4 block. For instance in the following N=2,3 are pushed to the left, and N=6 through N=11 shift down and squashes up horizontally.
DragonRounded DragonMidpoint 9--8 / \ 10 7 9---8 | | | | 11 6 10 7 / \ | | 5--4 <=> -11 6---5---4 \ | 3 3 | | 2 2 / | . 0--1 0---1
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::DragonRounded->new ()
$path = Math::PlanePath::DragonRounded->new (arms => $aa)
Create and return a new path object.
The optional arms
parameter makes a multi-arm curve. The default is 1 for just one arm.
($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 = $path->n_start()
Return 0, the first N in the path.
($n_lo, $n_hi) = $path->level_to_n_range($level)
Return (0, 2 * 2**$level - 1)
, or for multiple arms return (0, $arms * 2 * 2**$level - 1)
.
There are 2^level segments comprising the dragon, or arms*2^level when multiple arms. Each has 2 points in this rounded curve, numbered starting from 0.
The correspondence with the DragonMidpoint
noted above allows the method from that module to be used for the rounded xy_to_n()
.
The correspondence essentially reckons each point on the rounded curve as the midpoint of a dragon curve of one greater level of detail, and segments on 45-degree angles.
The coordinate conversion turns each 6x6 block of DragonRounded
to a 4x4 block of DragonMidpoint
. There's no rotations or anything.
Xmid = X - floor(X/3) - Xadj[X%6][Y%6] Ymid = Y - floor(Y/3) - Yadj[X%6][Y%6] N = DragonMidpoint n_to_xy of Xmid,Ymid Xadj[][] is a 6x6 table of 0 or 1 or undef Yadj[][] is a 6x6 table of -1 or 0 or undef
The Xadj,Yadj tables are a handy place to notice X,Y points not on the DragonRounded
style 4 of 9 points. Or 16 of 36 points since the tables are 6x6.
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include the various DragonCurve
sequences at even N, and in addition
http://oeis.org/A152822 (etc)
A152822 abs(dX), so 0=vertical,1=not, being 1,1,0,1 repeating A166486 abs(dY), so 0=horizontal,1=not, being 0,1,1,1 repeating
Math::PlanePath, Math::PlanePath::DragonCurve, Math::PlanePath::DragonMidpoint, Math::PlanePath::TerdragonRounded
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012, 2013, 2014, 2015 Kevin Ryde
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.
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