1Math::PlanePath::AlternUasteerPaCpoenrtMriidbpuotieMndatt(Ph3e:)r:lPlDaonceuPmaetnht:a:tAilotnernatePaperMidpoint(3)
2
6 Math::PlanePath::AlternatePaperMidpoint -- alternate paper folding
7 midpoints
8
10 use Math::PlanePath::AlternatePaperMidpoint;
11 my $path = Math::PlanePath::AlternatePaperMidpoint->new;
12 my ($x, $y) = $path->n_to_xy (123);
13
15 This is the midpoints of each alternate paper folding curve
16 (Math::PlanePath::AlternatePaper).
17 8 | 64-65-...
19 | |
20 7 | 63
21 | |
22 6 | 20-21 62
23 | | | |
24 5 | 19 22 61-60-59
25 | | | |
26 4 | 16-17-18 23 56-57-58
27 | | | |
28 3 | 15 26-25-24 55 50-49-48-47
29 | | | | | |
30 2 | 4--5 14 27-28-29 54 51 36-37 46
31 | | | | | | | | | |
32 1 | 3 6 13-12-11 30 53-52 35 38 45-44-43
33 | | | | | | | |
34 Y=0 | 0--1--2 7--8--9-10 31-32-33-34 39-40-41-42
35 +----------------------------------------------
36 X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
37 The "AlternatePaper" curve begins as follows and the midpoints are
39 numbered from 0,
40 |
42 9
43 |
44 --8--
45 | |
46 7 |
47 | |
48 --2-- --6--
49 | | |
50 1 3 5
51 | | |
52 *--0-- --4--
53 These midpoints are on fractions X=0.5,Y=0, X=1,Y=0.5, etc. For this
55 "AlternatePaperMidpoint" they're turned 45 degrees and mirrored so the
56 0,1,2 upward diagonal becomes horizontal along the X axis, and the
57 2,3,4 downward diagonal becomes a vertical at X=2, extending to X=2,Y=2
58 at N=4.
59 The midpoints are distinct X,Y positions because the alternate paper
61 curve traverses each edge only once.
62 The curve is self-similar in 2^level sections due to its unfolding.
64 This can be seen in the midpoints as for example N=0 to N=16 above is
65 the same shape as N=16 to N=32, but the latter rotated +90 degrees and
66 numbered in reverse.
67 Arms
69 The midpoints fill an eighth of the plane and eight copies can mesh
70 together perfectly when mirrored and rotated by 90, 180 and 270
71 degrees. The "arms" parameter can choose 1 to 8 curve arms
72 successively advancing.
73 For example "arms => 8" begins as follows. N=0,8,16,24,etc is the
75 first arm, the same as the plain curve above. N=1,9,17,25 is the
76 second, N=2,10,18,26 the third, etc.
77 90-82 81-89 7
79 arms => 8 | | | |
80 ... 74 73 ... 6
81 | |
82 66 65 5
83 | |
84 43-35 42-50-58 57-49-41 4
85 | | | |
86 91-.. 51 27 34-26-18 17-25-33 3
87 | | | | |
88 83-75-67-59 19-11--3 10 9 32-40 2
89 | | | |
90 84-76-68-60 20-12--4 2 1 24 48 ..-88 1
91 | | | | | |
92 92-.. 52 28 5 6 0--8-16 56-64-72-80 <- Y=0
93 | | | |
94 44-36 13 14 7-15-23 63-71-79-87 -1
95 | | | | |
96 37-29-21 22-30-38 31 55 ..-95 -2
97 | | | |
98 45-53-61 62-54-46 39-47 -3
99 | |
100 69 70 -4
101 | |
102 ... 77 78 ... -5
103 | | | |
104 93-85 86-94 -6
105 ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
107 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6
108 With eight arms like this every X,Y point is visited exactly once,
110 because the 8-arm "AlternatePaper" traverses every edge exactly once
111 ("Arms" in Math::PlanePath::AlternatePaper).
112 The arm numbering doesn't correspond to the "AlternatePaper", due to
114 the rotate and reflect of the first arm. It ends up arms 0 and 1 of
115 the "AlternatePaper" corresponding to arms 7 and 0 of the midpoints
116 here, those two being a pair going horizontally corresponding to a pair
117 in the "AlternatePaper" going diagonally into a quadrant.
118
120 See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
121 classes.
122 "$path = Math::PlanePath::AlternatePaperMidpoint->new ()"
124 Create and return a new path object.
125 "($x,$y) = $path->n_to_xy ($n)"
127 Return the X,Y coordinates of point number $n on the path. Points
128 begin at 0 and if "$n < 0" then the return is an empty list.
129 Fractional positions give an X,Y position along a straight line
131 between the integer positions.
132 "$n = $path->n_start()"
134 Return 0, the first N in the path.
135 Level Methods
137 "($n_lo, $n_hi) = $path->level_to_n_range($level)"
138 Return "(0, 2**$level - 1)", or for multiple arms return "(0, $arms
139 * (2**$level - 1)*$arms)". This is the same as the
140 "DragonMidpoint".
141
143 Entries in Sloane's Online Encyclopedia of Integer Sequences related to
144 this path include
145 <http://oeis.org/A016116> (etc)
147 A334576 X coordinate
149 A334577 Y coordinate
150 A016116 X/2 at N=2^k, being X/2=2^floor(k/2)
151
153 Math::PlanePath, Math::PlanePath::AlternatePaper
154 Math::PlanePath::DragonMidpoint, Math::PlanePath::R5DragonMidpoint,
156 Math::PlanePath::TerdragonMidpoint
157
159 <http://user42.tuxfamily.org/math-planepath/index.html>
160
162 Copyright 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin
163 Ryde
164 Math-PlanePath is free software; you can redistribute it and/or modify
166 it under the terms of the GNU General Public License as published by
167 the Free Software Foundation; either version 3, or (at your option) any
168 later version.
169 Math-PlanePath is distributed in the hope that it will be useful, but
171 WITHOUT ANY WARRANTY; without even the implied warranty of
172 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
173 General Public License for more details.
174 You should have received a copy of the GNU General Public License along
176 with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
177perl v5.36.0 202M3a-t0h1:-:2P0lanePath::AlternatePaperMidpoint(3)