1Math::PlanePath::TerdraUgsoenrMiCdopnotirnitb(u3t)ed PerMlatDho:c:uPmleannteaPtaitohn::TerdragonMidpoint(3)
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NAME

6       Math::PlanePath::TerdragonMidpoint -- dragon curve midpoints
7

SYNOPSIS

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

DESCRIPTION

14       This is midpoints of an integer version of the terdragon curve by Davis
15       and Knuth.
16
17                             30----29----28----27                      13
18                               \              /
19                                31          26                         12
20                                  \        /
21           36----35----34----33----32    25                            11
22             \                          /
23              37          41          24                               10
24                \        /  \        /
25                 38    40    42    23----22----21                       9
26                   \  /        \              /
27                    39          43          20                          8
28                                  \        /
29           48----47----46----45----44    19    12----11----10-----9     7
30             \                          /        \              /
31              49                      18          13           8        6
32                \                    /              \        /
33           ...---50                17----16----15----14     7           5
34                                                          /
35                                                         6              4
36                                                       /
37                                                      5-----4-----3     3
38                                                                /
39                                                               2        2
40                                                             /
41                                                            1           1
42                                                          /
43                                                         0         <- Y=0
44
45               ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^
46             -12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5 ...
47
48       The points are the middle of each edge of a double-size
49       "TerdragonCurve".
50
51                                   ...
52                                     \
53             6             -----8-----      double size
54                           \                TerdragonCurve
55                            \               giving midpoints
56             5               7
57                              \
58                               \
59             4        -----6---- _
60                      \         / \
61                       \       /   \
62             3          5     4     3
63                         \   /       \
64                          \_/         \
65             2              _----2-----
66                            \
67                             \
68             1                1
69                               \
70                                \
71           Y=0 ->    +-----0-----.
72
73                     ^
74                    X=0 1  2  3  4  5  6
75
76       For example in the "TerdragonCurve" N=3 to N=4 is X=3,Y=1 to X=2,Y=2
77       and that's doubled out here to X=6,Y=2 and X=4,Y=4 then the midpoint of
78       those positions is X=5,Y=3 for N=3 in the "TerdragonMidpoint".
79
80       The result is integer X,Y coordinates on every second point per
81       "Triangular Lattice" in Math::PlanePath, but visiting only 3 of every 4
82       such triangular points, which in turn is 3 of 8 all integer X,Y points.
83       The points used are a pattern of alternate rows with 1 of 2 points and
84       1 of 4 points.  For example the Y=7 row is 1 of 2 and the Y=8 row is 1
85       of 4.  Notice the pattern is the same when turned by 60 degrees.
86
87           * * * * * * * * * * * * * * * * * * * *
88            *   *   *   *   *   *   *   *   *   *
89           * * * * * * * * * * * * * * * * * * * *
90              *   *   *   *   *   *   *   *   *
91           * * * * * * * * * * * * * * * * * * * *
92            *   *   *   *   *   *   *   *   *   *
93           * * * * * * * * * * * * * * * * * * * *
94              *   *   *   *   *   *   *   *   *
95           * * * * * * * * * * * * * * * * * * * *
96            *   *   *   *   *   *   *   *   *   *
97           * * * * * * * * * * * * * * * * * * * *
98              *   *   *   *   *   *   *   *   *
99           * * * * * * * * * * * * * * * * * * * *
100            *   *   *   *   *   *   *   *   *   *
101           * * * * * * * * * * * * * * * * * * * *
102
103   Arms
104       Multiple copies of the curve can be selected, each advancing
105       successively.  Like the main "TerdragonCurve" the midpoint curve covers
106       1/6 of the plane and 6 arms rotated by 60, 120, 180, 240 and 300
107       degrees mesh together perfectly.  With 6 arms all the alternating
108       "1of2" and "1of4" points described above are visited.
109
110       "arms => 6" begins as follows.  N=0,6,12,18,etc is the first arm (like
111       the single curve above), then N=1,7,13,19 the second copy rotated 60
112       degrees, N=2,8,14,20 the third rotated 120, etc.
113
114            arms=>6                                 ...
115                                                    /
116                    ...                           42
117                      \                          /
118                       43          19          36
119                         \        /  \        /
120                          37    25    13    30----24----18
121                            \  /        \              /
122                             31           7          12
123                                           \        /
124                    20----14-----8-----2     1     6    35----41----47-..
125                      \                          /        \
126                       26           3     .     0          29
127                         \        /                          \
128           ..-44----38----32     9     4     5----11----17----23
129                               /        \
130                             15          10          34
131                            /              \        /  \
132                          21----27----33    16    28    40
133                                     /        \  /        \
134                                   39          22          46
135                                  /                          \
136                                45                            ...
137                               /
138                             ...
139

FUNCTIONS

141       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
142       classes.
143
144       "$path = Math::PlanePath::TerdragonMidpoint->new ()"
145           Create and return a new path object.
146
147       "($x,$y) = $path->n_to_xy ($n)"
148           Return the X,Y coordinates of point number $n on the path.  Points
149           begin at 0 and if "$n < 0" then the return is an empty list.
150
151           Fractional positions give an X,Y position along a straight line
152           between the integer positions.
153
154       "$n = $path->n_start()"
155           Return 0, the first N in the path.
156
157   Level Methods
158       "($n_lo, $n_hi) = $path->level_to_n_range($level)"
159           Return "(0, 3**$level - 1)", or for multiple arms return "(0, $arms
160           * 3**$level - 1)".
161
162           There are 3^level segments comprising the terdragon, or
163           arms*3^level when multiple arms, numbered starting from 0.
164

FORMULAS

166   X,Y to N
167       An X,Y point can be turned into N by dividing out digits of a complex
168       base b=w+1 where
169
170           w = 1/2 + i * sqrt(3)/2            w^2     w
171             = 6th root of unity                 \   /
172                                                  \ /
173                                       w^3=-1 -----o------ w^0=1
174                                                  / \
175                                                 /   \
176                                              w^4     w^5
177
178       At each step the low ternary digit is formed from X,Y and an adjustment
179       applied to move X,Y onto a multiple of w+1 ready to divide out w+1.
180
181       In the N points above it can be seen that each group of three N values
182       make a straight line, such as N=0,1,2, or N=3,4,5 etc.  The adjustment
183       moves the two ends N=0mod3 or N=2mod3 to the centre N=1mod3.  The
184       centre N=1mod3 position is always a multiple of w+1.
185
186       The angles and positions for the N triples follow a 12-point pattern as
187       follows, where each / \ or - is a point on the path (any arm).
188
189            \   /   /   \   /   /   \   /   /   \   /   /   \
190           - \ / \ - - - \ / \ - - - \ / \ - - - \ / \ - - -
191              /   \   /   /   \   /   /   \   /   /   \   /
192           \ - - - \ / \ - - - \ / \ - - - \ / \ - - - \ / \
193            \   /   /   \   /   /   \   /   /   \   /   /   \
194           - \ / \ - - - \ / \ - - - \ / \ - - - \ / \ - - -
195              /   \   /   /   \   /   /   \   /   /   \   /
196           \ - - - \ / \ - - - \ / \ - - - \ / \ - - - \ / \
197            \   /   /   \   /   /   \   /   /   \   /   /   \
198           - \ / \ - - - \ / \ - - - \ / \ - - - \ / \ - - -
199              /   \   /   /   \   /   /   \   /   /   \   /
200           \ - - - \ / \ - - - \ / \ - - - \ / \ - - - \ / \
201            \   /   /   \   /   /   \   /   /   \   /   /   \
202           - \ / \ - - - \ / \ - - - \ / \ - - - \ / \ - - -
203              /   \   /   /   \   /   /   \   /   /   \   /
204           \ - - - \ / \ - - - \ / \ - - - \ / \ - - - \ / \
205            \   /   /   \   /   /   \   /   /   \   /   /   \
206           - \ / \ - - - \ / \ - - - \ / \ - - - \ / \ - - -
207              /   \   /   /   \   /   /   \   /   /   \   /
208           \ - - - \ / \ - - - \ / \ - - - \ / \ - - - \ / \
209            \   /   /   \   /   /   \   /   /   \   /   /   \
210           - \ / \ - - - \ / \ - - - \ / \ - - - \ / \ - - -
211              /   \   /   /   \   /   /   \   /   /   \   /
212           \ - - - \ / \ - - - \ / \ - - - \ / \ - - - \ / \
213
214       In the current code a 12x12 table is used, indexed by X mod 12 and Y
215       mod 12.  With Xadj and Yadj from there
216
217           Ndigit = (X + 1) mod 3      # N digits low to high
218
219           Xm = X + Xadj[X mod 12, Y mod 12]
220           Ym = Y + Yadj[X mod 12, Y mod 12]
221
222           new X,Y = (Xm,Ym) / (w+1)
223                   = (Xm,Ym) * (2-w) / 3
224                   = ((Xm+Ym)/2, (Ym-(Xm/3))/2)
225
226       Is there a good aX+bY mod 12 or mod 24 for a smaller table?  Maybe X+3Y
227       like the digit?  Taking C=(X-Y)/2 in triangular coordinate style can
228       reduce the table to 6x6.
229
230       Points not reached by the curve (ie. not the 3 of 4 triangular or 3 of
231       8 rectangular described above) can be detected with "undef" or suitably
232       tagged entries in the adjustment table.
233
234       The X,Y reduction stops at the midpoint of the first triple of the
235       originating arm.  So X=3,Y=1 which is N=1 for the first arm, and that
236       point rotated by 60,120,180,240,300 degrees for the others.  If only
237       some of the arms are of interest then reaching one of the others means
238       the original X,Y was outside the desired region.
239
240           Arm     X,Y Endpoint
241           ---     ------------
242            0        3,1
243            1        0,2
244            2       -3,1
245            3       -3,-1
246            4        0,-2
247            5        3,-1
248
249       For the odd arms 1,3,5 each digit of N must be flipped 2-digit so 0,1,2
250       becomes 2,1,0,
251
252           if arm odd
253           then  N = 3**numdigits - 1 - N
254

SEE ALSO

256       Math::PlanePath, Math::PlanePath::TerdragonCurve,
257       Math::PlanePath::TerdragonRounded
258
259       Math::PlanePath::DragonMidpoint, Math::PlanePath::R5DragonMidpoint
260

HOME PAGE

262       <http://user42.tuxfamily.org/math-planepath/index.html>
263

LICENSE

265       Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020
266       Kevin Ryde
267
268       This file is part of Math-PlanePath.
269
270       Math-PlanePath is free software; you can redistribute it and/or modify
271       it under the terms of the GNU General Public License as published by
272       the Free Software Foundation; either version 3, or (at your option) any
273       later version.
274
275       Math-PlanePath is distributed in the hope that it will be useful, but
276       WITHOUT ANY WARRANTY; without even the implied warranty of
277       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
278       General Public License for more details.
279
280       You should have received a copy of the GNU General Public License along
281       with Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.
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285perl v5.36.0                      2022-07-M2a2th::PlanePath::TerdragonMidpoint(3)
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