1Math::PlanePath::KochelUCsuerrveC(o3n)tributed Perl DocuMmaetnht:a:tPiloannePath::KochelCurve(3)
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NAME

6       Math::PlanePath::KochelCurve -- 3x3 self-similar R and F
7

SYNOPSIS

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

DESCRIPTION

14       This is an integer version of the Kochel curve by Herman Haverkort.  It
15       fills the first quadrant in a 3x3 self-similar pattern made from two
16       base shapes.
17
18                   |
19             8    80--79  72--71--70--69  60--59--58
20                       |   |           |   |       |
21             7    77--78  73  66--67--68  61  56--57
22                   |       |   |           |   |
23             6    76--75--74  65--64--63--62  55--54
24                                                   |
25             5    11--12  17--18--19--20  47--48  53
26                   |   |   |           |   |   |   |
27             4    10  13  16  25--24  21  46  49  52
28                   |   |   |   |   |   |   |   |   |
29             3     9  14--15  26  23--22  45  50--51
30                   |           |           |
31             2     8-- 7-- 6  27--28--29  44--43--42
32                           |           |           |
33             1     1-- 2   5  32--31--30  37--38  41
34                   |   |   |   |           |   |   |
35           Y=0->   0   3-- 4  33--34--35--36  39--40
36
37                   X=0  1   2   3   4   5   6   7   8   9  10  11  12  13  14
38
39       The base shapes are an "R" and an "F".  The R goes along an edge, the F
40       goes diagonally across.
41
42                 R pattern                      F pattern   ^
43           +------+-----+-----+           +------+-----+----|+
44           |2   | |3\   |4    |           |2   | |3\   |8   ||
45           |  R | |  F  |   R |           |  R | |  F  |  R ||
46           |    | |   \ |-----|           |    | |   \ |    ||
47           +------+-----+-----+           +------+-----+-----+
48           |1  /  |6    |5  / |           |1  /  |4  / |7  / |
49           |  F   | Rrev|  F  |           |  F   |  F  |  F  |
50           | /    |-----| /   |           | /    | /   | /   |
51           +------+-----+-----+           +------+-----+-----+
52           |0|    |7\   |8    |           |0|    |5\   ||6   |
53           | |Rrev|  F  |  R  |           | |Rrev|  F  ||Rrev|
54           | o    |   \ |------>          | o    |   \ ||    |
55           +------+-----+-----+           +------+-----+-----+
56
57       "Rrev" means the R pattern followed in reverse, which is
58
59           +------+-----+-----+
60           |8<----|7\   |6    |    Rrev pattern
61           |   R  |  F  | Rrev|
62           |      |   \ |-----|    turned -90 degrees
63           +------+-----+-----+    so as to start at
64           |1  /  ||2   |5  / |    bottom left
65           |  F   || R  |  F  |
66           | /    ||    | /   |
67           +------+-----+-----+
68           |0|    |3\   ||4   |
69           | |Rrev|  F  ||Rrev|
70           | o    |   \ ||    |
71           +------+-----+-----+
72
73       The F pattern is symmetric, the same forward or reverse, including its
74       sub-parts taken in reverse, so there's no separate "Frev" pattern.
75
76       The initial N=0 to N=8 is the Rrev turned -90, then N=9 to N=17 is the
77       F shape.  The next higher level N=0,N=9,N=18 to N=72 is the Rrev too,
78       as is any N=9^k to N=8*9^k.
79
80   Fractal
81       The curve is conceived by Haverkort for filling a unit square by
82       descending into ever-smaller replacements, like other space-filling
83       curves.  For that the top-level can be any of the patterns.  To descend
84       any of the shapes can be used for the start, but for the outward
85       expanding version here the starting pattern must occur at the start of
86       its next higher level, which means Rrev is the only choice as it's the
87       only start in any of the three patterns.
88
89       But all the patterns can be found in the path at any desired size.  For
90       example the "1" part of Rrev is an F, which means F to a desired level
91       can be found at
92
93           NFstart = 1 * 9^level
94           NFlast = NFstart + 9^level - 1
95                  = 2 * 9^level - 1
96           XFstart = 3^level
97           YFstart = 0
98
99       level=3 for N=729 to N=1457 is a 27x27 which is the level-three F shown
100       in Haverkort's paper, starting at XFstart=27,YFstart=0.
101

FUNCTIONS

103       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
104       classes.
105
106       "$path = Math::PlanePath::KochelCurve->new ()"
107           Create and return a new path object.
108
109       "($x,$y) = $path->n_to_xy ($n)"
110           Return the X,Y coordinates of point number $n on the path.  Points
111           begin at 0 and if "$n < 0" then the return is an empty list.
112
113   Level Methods
114       "($n_lo, $n_hi) = $path->level_to_n_range($level)"
115           Return "(0, 9**$level - 1)".
116

SEE ALSO

118       Math::PlanePath, Math::PlanePath::PeanoCurve,
119       Math::PlanePath::WunderlichMeander
120
121       Herman Haverkort, "Recursive Tilings and Space-Filling Curves with
122       Little Fragmentation", Journal of Computational Geometry, 2(1), 92-127,
123       2011.
124
125           <http://jocg.org/index.php/jocg/article/view/68>
126           <http://jocg.org/index.php/jocg/article/download/68/20>
127           <http://arxiv.org/abs/1002.1843>
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129           <http://alexandria.tue.nl/openaccess/Metis239505.pdf> (slides)
130           <http://www.win.tue.nl/~hermanh/stack/h-rtslf-eurocg2010-talk.pdf>
131           (short form)
132

HOME PAGE

134       <http://user42.tuxfamily.org/math-planepath/index.html>
135

LICENSE

137       Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017 Kevin Ryde
138
139       This file is part of Math-PlanePath.
140
141       Math-PlanePath is free software; you can redistribute it and/or modify
142       it under the terms of the GNU General Public License as published by
143       the Free Software Foundation; either version 3, or (at your option) any
144       later version.
145
146       Math-PlanePath is distributed in the hope that it will be useful, but
147       WITHOUT ANY WARRANTY; without even the implied warranty of
148       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
149       General Public License for more details.
150
151       You should have received a copy of the GNU General Public License along
152       with Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.
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156perl v5.28.0                      2017-12-03   Math::PlanePath::KochelCurve(3)
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