1Math::PlanePath::AR2W2CUusrevre(C3o)ntributed Perl DocumMeanttha:t:iPolnanePath::AR2W2Curve(3)
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6 Math::PlanePath::AR2W2Curve -- 2x2 self-similar curve of four patterns
7
9 use Math::PlanePath::AR2W2Curve;
10 my $path = Math::PlanePath::AR2W2Curve->new;
11 my ($x, $y) = $path->n_to_xy (123);
12
14 This is an integer version of the AR2W2 curve per
15 Asano, Ranjan, Roos, Welzl and Widmayer "Space-Filling Curves and
17 Their Use in the Design of Geometric Data Structures", Theoretical
18 Computer Science, volume 181, issue 1, pages 3-15, July 1997.
19 And in LATIN'95 Theoretical Informatics which is at Google Books
21 <http://books.google.com.au/books?id=_aKhJUJunYwC&pg=PA36>
22 It traverses the first quadrant in self-similar 2x2 blocks which are a
24 mixture of "U" and "Z" shapes. The mixture is designed to improve some
25 locality measures (how big the N range for a given region).
26 |
28 7 42--43--44 47--48--49 62--63
29 \ | | | |
30 6 40--41 45--46 51--50 61--60
31 | | |
32 5 39 36--35--34 52 55--56 59
33 | | / | | | |
34 4 38--37 33--32 53--54 57--58
35 \
36 3 6-- 7-- 8 10 31 28--27--26
37 | |/ | | | |
38 2 5-- 4 9 11 30--29 24--25
39 | | |
40 1 2-- 3 13--12 17--18 23--22
41 \ | | | |
42 Y=0 -> 0-- 1 14--15--16 19--20--21
43 X=0 1 2 3 4 5 6 7
45 Shape Parts
47 There's four base patterns A to D. A2 is a mirror image of A1, B2 a
48 mirror of B1, etc. The start is A1, and above that D2, then A1 again,
49 alternately.
50 ^----> ^
52 2---3 C1 | B2 1 3 C2 D1 |
53 A1 \ | A2 | \ | ----> |
54 0---1 ^ 0 2 ^ ---->
55 D2 | B1 |B1 B2
56 ---->| |
57 1---2 C2 B1 1---2 B2 C1
60 B1 | | ---->----> B2 | | ---->---->
61 0 3 ^ | 0 3 ^ |
62 |D1 B2| |B1 D2|
63 | v | v
64 ^ \ ^ |
66 1---2 B1| \A1 1---2 A2/ | B2
67 C1 | | | v C2 | | / v
68 0 3 ^ | 0 3 ^ \
69 /A2 B2| |B1 \A1
70 / v | v
71 ^ | ^ \
73 1---2 A2/ | C2 1---2 C1| \A1
74 D1 | | / v D2 | | | v
75 0 3 ^ \ 0 3 ^ |
76 |D1 \A2 /A1 D2|
77 | v / v
78 For parts which fill on the right such as the B1 and B2 sub-parts of
80 A1, the numbering must be reversed. This doesn't affect the shape of
81 the curve as such, but it matters for enumerating it as done here.
82 Start Shape
84 The default starting shape is the A1 "Z" part, and above it D2. Notice
85 the starting sub-part of D2 is A1 and in turn the starting sub-part of
86 A1 is D2, so those two alternate at successive higher levels. Their
87 sub-parts reach all other parts (in all directions, and forward or
88 reverse).
89 The "start_shape => $str" option can select a different starting shape.
91 The choices are
92 "A1" \ pair
94 "D2" /
95 "B2" \ pair
96 "B1rev" /
97 "D1rev" \ pair
98 "A2rev" /
99 B2 begins with a reversed B1 and in turn a B1 reverse begins with B2
101 (no reverse), so those two alternate. Similarly D1 reverse starts with
102 A2 reverse, and A2 reverse starts with D1 reverse.
103 The curve is conceived by the authors as descending into ever-smaller
105 sub-parts and for that any of the patterns can be a top-level start.
106 But to expand outwards as done here the starting part must be the start
107 of the pattern above it, and that's so only for the 6 listed. The
108 descent graph is
109 D2rev -----> D2 <--> A1
111 B2rev ----->
112 C2rev --> A1rev -----> B2 <--> B1rev <----- C2
114 C1rev -----> <----- A2 <-- C1
115 B1 -----> D1rev <--> A2rev
117 D1 ----->
118 So for example B1 is not at the start of anything. Or A1rev is at the
120 start of C2rev, but then nothing starts with C2rev. Of the 16 total
121 only the three pairs shown "<-->" are cycles and can thus extend
122 upwards indefinitely.
123
125 See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
126 classes.
127 "$path = Math::PlanePath::AR2W2Curve->new ()"
129 Create and return a new path object.
130 "($x,$y) = $path->n_to_xy ($n)"
132 Return the X,Y coordinates of point number $n on the path. Points
133 begin at 0 and if "$n < 0" then the return is an empty list.
134 "($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)"
136 The returned range is exact, meaning $n_lo and $n_hi are the
137 smallest and largest in the rectangle.
138 Level Methods
140 "($n_lo, $n_hi) = $path->level_to_n_range($level)"
141 Return "(0, 4**$level - 1)".
142
144 Math::PlanePath, Math::PlanePath::HilbertCurve,
145 Math::PlanePath::PeanoCurve
146
148 <http://user42.tuxfamily.org/math-planepath/index.html>
149
151 Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020
152 Kevin Ryde
153 This file is part of Math-PlanePath.
155 Math-PlanePath is free software; you can redistribute it and/or modify
157 it under the terms of the GNU General Public License as published by
158 the Free Software Foundation; either version 3, or (at your option) any
159 later version.
160 Math-PlanePath is distributed in the hope that it will be useful, but
162 WITHOUT ANY WARRANTY; without even the implied warranty of
163 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
164 General Public License for more details.
165 You should have received a copy of the GNU General Public License along
167 with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
168perl v5.34.0 2022-01-21 Math::PlanePath::AR2W2Curve(3)