1Math::PlanePath::DragonURsoeurndCeodn(t3r)ibuted Perl DoMcautmhe:n:tPaltainoenPath::DragonRounded(3)
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NAME

6       Math::PlanePath::DragonRounded -- dragon curve, with rounded corners
7

SYNOPSIS

9        use Math::PlanePath::DragonRounded;
10        my $path = Math::PlanePath::DragonRounded->new;
11        my ($x, $y) = $path->n_to_xy (123);
12

DESCRIPTION

14       This is a version of the dragon curve by Harter, Heighway, et al, done
15       with two points per edge and skipping vertices so as to make rounded-
16       off corners,
17
18                                 17-16              9--8                 6
19                                /     \           /     \
20                              18       15       10        7              5
21                               |        |        |        |
22                              19       14       11        6              4
23                                \        \     /           \
24                                 20-21    13-12              5--4        3
25                                      \                          \
26                                       22                          3     2
27                                        |                          |
28                                       23                          2     1
29                                      /                          /
30               33-32             25-24                    .  0--1       Y=0
31              /     \           /
32            34       31       26                                        -1
33             |        |        |
34            35       30       27                                        -2
35              \        \     /
36               36-37    29-28    44-45                                  -3
37                    \           /     \
38                     38       43       46                               -4
39                      |        |        |
40                     39       42       47                               -5
41                       \     /        /
42                        40-41    49-48                                  -6
43                                /
44                              50                                        -7
45                               |
46                              ...
47
48
49             ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^
50           -15-14-13-12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3 ...
51
52       The two points on an edge have one of X or Y a multiple of 3 and the
53       other Y or X at 1 mod 3 or 2 mod 3.  For example N=19 and N=20 are on
54       the X=-9 edge (a multiple of 3), and at Y=4 and Y=5 (1 and 2 mod 3).
55
56       The "rounding" of the corners ensures that for example N=13 and N=21
57       don't touch as they approach X=-6,Y=3.  The curve always approaches
58       vertices like this and never crosses itself.
59
60   Arms
61       The dragon curve fills a quarter of the plane and four copies mesh
62       together rotated by 90, 180 and 270 degrees.  The "arms" parameter can
63       choose 1 to 4 curve arms, successively advancing.  For example "arms =>
64       4" gives
65
66                       36-32             59-...          6
67                      /     \           /
68           ...      40       28       55                 5
69            |        |        |        |
70           56       44       24       51                 4
71             \     /           \        \
72              52-48    13--9    20-16    47-43           3
73                      /     \        \        \
74                    17        5       12       39        2
75                     |        |        |        |
76                    21        1        8       35        1
77                   /                 /        /
78              29-25     6--2     0--4    27-31       <- Y=0
79             /        /                 /
80           33       10        3       23                -1
81            |        |        |        |
82           37       14        7       19                -2
83             \        \        \     /
84              41-45    18-22    11-15    50-54          -3
85                   \        \           /     \
86                    49       26       46       58       -4
87                     |        |        |        |
88                    53       30       42       ...      -5
89                   /           \     /
90             ...-57             34-38                   -6
91
92
93
94            ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^
95           -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6
96
97       With 4 arms like this all 3x3 blocks are visited, using 4 out of 9
98       points in each.
99
100   Midpoint
101       The points of this rounded curve correspond to the "DragonMidpoint"
102       with a little squish to turn each 6x6 block into a 4x4 block.  For
103       instance in the following N=2,3 are pushed to the left, and N=6 through
104       N=11 shift down and squashes up horizontally.
105
106            DragonRounded               DragonMidpoint
107
108               9--8
109              /    \
110            10      7                     9---8
111             |      |                     |   |
112            11      6                    10   7
113           /         \                    |   |
114                      5--4      <=>     -11   6---5---4
115                          \                           |
116                           3                          3
117                           |                          |
118                           2                          2
119                          /                           |
120                    . 0--1                        0---1
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FUNCTIONS

123       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
124       classes.
125
126       "$path = Math::PlanePath::DragonRounded->new ()"
127       "$path = Math::PlanePath::DragonRounded->new (arms => $aa)"
128           Create and return a new path object.
129
130           The optional "arms" parameter makes a multi-arm curve.  The default
131           is 1 for just one arm.
132
133       "($x,$y) = $path->n_to_xy ($n)"
134           Return the X,Y coordinates of point number $n on the path.  Points
135           begin at 0 and if "$n < 0" then the return is an empty list.
136
137       "$n = $path->n_start()"
138           Return 0, the first N in the path.
139
140   Level Methods
141       "($n_lo, $n_hi) = $path->level_to_n_range($level)"
142           Return "(0, 2 * 2**$level - 1)", or for multiple arms return "(0,
143           $arms * 2 * 2**$level - 1)".
144
145           There are 2^level segments comprising the dragon, or arms*2^level
146           when multiple arms.  Each has 2 points in this rounded curve,
147           numbered starting from 0.
148

FORMULAS

150   X,Y to N
151       The correspondence with the "DragonMidpoint" noted above allows the
152       method from that module to be used for the rounded "xy_to_n()".
153
154       The correspondence essentially reckons each point on the rounded curve
155       as the midpoint of a dragon curve of one greater level of detail, and
156       segments on 45-degree angles.
157
158       The coordinate conversion turns each 6x6 block of "DragonRounded" to a
159       4x4 block of "DragonMidpoint".  There's no rotations or anything.
160
161           Xmid = X - floor(X/3) - Xadj[X%6][Y%6]
162           Ymid = Y - floor(Y/3) - Yadj[X%6][Y%6]
163
164           N = DragonMidpoint n_to_xy of Xmid,Ymid
165
166           Xadj[][] is a 6x6 table of 0 or 1 or undef
167           Yadj[][] is a 6x6 table of -1 or 0 or undef
168
169       The Xadj,Yadj tables are a handy place to notice X,Y points not on the
170       "DragonRounded" style 4 of 9 points.  Or 16 of 36 points since the
171       tables are 6x6.
172

OEIS

174       Entries in Sloane's Online Encyclopedia of Integer Sequences related to
175       this path include the various "DragonCurve" sequences at even N, and in
176       addition
177
178           <http://oeis.org/A152822> (etc)
179
180           A152822   abs(dX), so 0=vertical,1=not, being 1,1,0,1 repeating
181           A166486   abs(dY), so 0=horizontal,1=not, being 0,1,1,1 repeating
182

SEE ALSO

184       Math::PlanePath, Math::PlanePath::DragonCurve,
185       Math::PlanePath::DragonMidpoint, Math::PlanePath::TerdragonRounded
186

HOME PAGE

188       <http://user42.tuxfamily.org/math-planepath/index.html>
189

LICENSE

191       Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020
192       Kevin Ryde
193
194       Math-PlanePath is free software; you can redistribute it and/or modify
195       it under the terms of the GNU General Public License as published by
196       the Free Software Foundation; either version 3, or (at your option) any
197       later version.
198
199       Math-PlanePath is distributed in the hope that it will be useful, but
200       WITHOUT ANY WARRANTY; without even the implied warranty of
201       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
202       General Public License for more details.
203
204       You should have received a copy of the GNU General Public License along
205       with Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.
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209perl v5.34.0                      2021-07-22 Math::PlanePath::DragonRounded(3)
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