NAME
Math::PlanePath::DragonRounded  dragon curve, with rounded corners
SYNOPSIS
use Math::PlanePath::DragonRounded;
my $path = Math::PlanePath::DragonRounded>new;
my ($x, $y) = $path>n_to_xy (123);
DESCRIPTION
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 roundedoff corners,
1716 98 6
/ \ / \
18 15 10 7 5
   
19 14 11 6 4
\ \ / \
2021 1312 54 3
\ \
22 3 2
 
23 2 1
/ /
3332 2524 . 01 Y=0
/ \ /
34 31 26 1
  
35 30 27 2
\ \ /
3637 2928 4445 3
\ / \
38 43 46 4
  
39 42 47 5
\ / /
4041 4948 6
/
50 7

...
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
151413121110 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.
Arms
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
3632 59... 6
/ \ /
... 40 28 55 5
   
56 44 24 51 4
\ / \ \
5248 139 2016 4743 3
/ \ \ \
17 5 12 39 2
   
21 1 8 35 1
/ / /
2925 62 04 2731 < Y=0
/ / /
33 10 3 23 1
   
37 14 7 19 2
\ \ \ /
4145 1822 1115 5054 3
\ \ / \
49 26 46 58 4
   
53 30 42 ... 5
/ \ /
...57 3438 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.
Midpoint
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
98
/ \
10 7 98
   
11 6 10 7
/ \  
54 <=> 11 654
\ 
3 3
 
2 2
/ 
. 01 01
FUNCTIONS
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 multiarm 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.
Level Methods
($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.
FORMULAS
X,Y to N
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 45degree 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.
OEIS
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
SEE ALSO
Math::PlanePath, Math::PlanePath::DragonCurve, Math::PlanePath::DragonMidpoint, Math::PlanePath::TerdragonRounded
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde
MathPlanePath 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.
MathPlanePath 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 MathPlanePath. If not, see <http://www.gnu.org/licenses/>.