Provided by: libmath-planepath-perl_129-1_all 
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 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.
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
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.
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
9--8
/ \
10 7 9---8
| | | |
11 6 10 7
/ \ | |
5--4 <=> -11 6---5---4
\ |
3 3
| |
2 2
/ |
. 0--1 0---1
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$path = Math::PlanePath::DragonRounded->new ()"
"$path = Math::PlanePath::DragonRounded->new (arms => $aa)"
Create and return a new path object.
The optional "arms" parameter makes a multi-arm curve. The default is 1 for just one
arm.
"($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 = $path->n_start()"
Return 0, the first N in the path.
Level Methods
"($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 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.
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/math-planepath/index.html>
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 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.
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/>.