1Math::PlanePath::WunderUlsiecrhSCeornpternitbiuntee(d3M)Paetrhl::DPolcaunmeePnattaht:i:oWnunderlichSerpentine(3)
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NAME

6       Math::PlanePath::WunderlichSerpentine -- transpose parts of Peano
7       curve, including coil order
8

SYNOPSIS

10        use Math::PlanePath::WunderlichSerpentine;
11        my $path = Math::PlanePath::WunderlichSerpentine->new (serpentine_type => '111_000_111');
12        my ($x, $y) = $path->n_to_xy (123);
13
14        # or another radix digits ...
15        my $path5 = Math::PlanePath::WunderlichSerpentine->new (radix => 5);
16

DESCRIPTION

18       This is an integer version of Walter Wunderlich's variations on the
19       "PeanoCurve".  A "serpentine type" controls transposing of selected 3x3
20       sub-parts.  The default is "alternating" 010,101,010 which transposes
21       every second sub-part,
22
23              8  | 60--61--62--63  68--69  78--79--80--81
24                 |  |           |   |   |   |           |
25              7  | 59--58--57  64  67  70  77--76--75  ...
26                 |          |   |   |   |           |
27              6  | 54--55--56  65--66  71--72--73--74
28                 |  |
29              5  | 53  48--47  38--37--36--35  30--29
30                 |  |   |   |   |           |   |   |
31              4  | 52  49  46  39--40--41  34  31  28
32                 |  |   |   |           |   |   |   |
33              3  | 51--50  45--44--43--42  33--32  27
34                 |                                  |
35              2  |  6-- 7-- 8-- 9  14--15  24--25--26
36                 |  |           |   |   |   |
37              1  |  5-- 4-- 3  10  13  16  23--22--21
38                 |          |   |   |   |           |
39             Y=0 |  0-- 1-- 2  11--12  17--18--19--20
40                 |
41                 +-------------------------------------
42                   X=0  1   2   3   4   5   6   7   8
43
44       "serpentine_type" can be a string of 0s and 1s, with optional space,
45       comma or _ separators at each group of 3,
46
47           "011111011"         0/1 string
48           "011,111,011"
49           "011_111_011"
50           "011 111 011"
51
52       or special values
53
54           "alternating"       01010101.. the default
55           "coil"              11111... all 1s described below
56           "Peano"             00000... all 0s, gives PeanoCurve
57
58       Each "1" sub-part is transposed.  The string is applied in order of the
59       N parts, irrespective of what net reversals and transposes are in force
60       on a particular part.
61
62       When no parts are transposed, which is a string of all 0s, the result
63       is the same as the "PeanoCurve".  The special "serpentine_type =>
64       "Peano"" gives that.
65
66   Coil Order
67       "serpentine_type => "coil"" means "111 111 111" to transpose all parts.
68       The result is like a coil viewed side-on,
69
70            8      24--25--26--27--28--29  78--79--80--81--...
71                    |                   |   |
72            7      23--22--21  32--31--30  77--76--75
73                            |   |                   |
74            6      18--19--20  33--34--35  72--73--74
75                    |                   |   |
76            5      17--16--15  38--37--36  71--70--69
77                            |   |                   |
78            4      12--13--14  39--40--41  66--67--68
79                    |                   |   |
80            3      11--10-- 9  44--43--42  65--64--63
81                            |   |                   |
82            2       6-- 7-- 8  45--46--47  60--61--62
83                    |                   |   |
84            1       5-- 4-- 3  50--49--48  59--58--57
85                            |   |                   |
86           Y=0      0-- 1-- 2  51--52--53--54--55--56
87
88                  X=0   1   2   3   4   5   6   7   8
89
90       Whenever "serpentine_type" begins with a "1" the initial sub-part is
91       transposed at each level.  The first step N=0 to N=1 is kept fixed
92       along the X axis, then the higher levels are transposed.  For example
93       in the coil above The N=9 to N=17 part is upwards, and then the N=81 to
94       N=161 part is to the right, and so on.
95
96   Radix
97       The optional "radix" parameter gives the size of the sub-parts, similar
98       to the "PeanoCurve" "radix" parameter (see "Radix" in
99       Math::PlanePath::PeanoCurve).  For example radix 5 gives
100
101            radix => 5
102
103            4  | 20-21-22-23-24-25 34-35 44-45 70-71-72-73-74-75 84-85
104               |  |              |  |  |  |  |  |              |  |  |
105            3  | 19-18-17-16-15 26 33 36 43 46 69-68-67-66-65 76 83 86
106               |              |  |  |  |  |  |              |  |  |  |
107            2  | 10-11-12-13-14 27 32 37 42 47 60-61-62-63-64 77 82 87
108               |  |              |  |  |  |  |  |              |  |  |
109            1  |  9--8--7--6--5 28 31 38 41 48 59-58-57-56-55 78 81 88
110               |              |  |  |  |  |  |              |  |  |  |
111           Y=0 |  0--1--2--3--4 29-30 39-40 49-50-51-52-53-54 79-80 89-..
112               +---------------------------------------------------------
113                X=0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
114
115       Like the "PeanoCurve" if the radix is even then the ends of each sub-
116       part don't join up.  For example in radix 4 N=15 isn't next to N=16,
117       nor N=31 to N=32, etc.
118
119               |                                                              |
120            3  | 15--14--13--12  16  23--24  31  47--46--45--44  48  55--56  63
121               |              |   |   |   |   |               |   |   |   |   |
122            2  |  8-- 9--10--11  17  22  25  30  40--41--42--43  49  54  57  62
123               |  |               |   |   |   |   |               |   |   |   |
124            1  |  7-- 6-- 5-- 4  18  21  26  29  39--38--37--36  50  53  58  61
125               |              |   |   |   |   |               |   |   |   |   |
126           Y=0 |  0-- 1-- 2-- 3  19--20  27--28  32--33--34--35  51--52  59--60
127               +----------------------------------------------------------------
128                 X=0  1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
129
130       In the "serpentine_type" 0,1 form, any space, comma, etc, separators
131       should group "radix" many values, so for example
132
133           serpentine_type => "00000_11111_00000_00000_11111"
134
135       The intention is to do something friendly if the separators are not on
136       such boundaries, so that say 000_111_000 can have a sensible meaning in
137       a radix higher than 3.  But exactly what is not settled, so always give
138       a full string of desired 0,1 for now.
139

FUNCTIONS

141       See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path
142       classes.
143
144       "$path = Math::PlanePath::WunderlichSerpentine->new ()"
145       "$path = Math::PlanePath::WunderlichSerpentine->new (serpentine_type =>
146       $str, radix => $r)"
147           Create and return a new path object.
148
149           The optional "radix" parameter gives the base for digit splitting.
150           The default is ternary, radix 3.  The radix should be an odd
151           number, 3, 5, 7, 9 etc.
152

SEE ALSO

154       Math::PlanePath, Math::PlanePath::PeanoCurve
155
156       Walter Wunderlich "Uber Peano-Kurven", Elemente der Mathematik,
157       28(1):1-10, 1973.
158       <http://sodwana.uni-ak.ac.at/geom/mitarbeiter/wallner/wunderlich/>
159       <http://sodwana.uni-ak.ac.at/geom/mitarbeiter/wallner/wunderlich/pdf/125.pdf>
160

HOME PAGE

162       <http://user42.tuxfamily.org/math-planepath/index.html>
163

LICENSE

165       Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017 Kevin Ryde
166
167       This file is part of Math-PlanePath.
168
169       Math-PlanePath is free software; you can redistribute it and/or modify
170       it under the terms of the GNU General Public License as published by
171       the Free Software Foundation; either version 3, or (at your option) any
172       later version.
173
174       Math-PlanePath is distributed in the hope that it will be useful, but
175       WITHOUT ANY WARRANTY; without even the implied warranty of
176       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
177       General Public License for more details.
178
179       You should have received a copy of the GNU General Public License along
180       with Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.
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184perl v5.28.0                      2017-M1a2t-h0:3:PlanePath::WunderlichSerpentine(3)
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