1Math::PlanePath::ImaginUasreyrHaClofn(t3r)ibuted Perl DoMcautmhe:n:tPaltainoenPath::ImaginaryHalf(3)
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NAME

6       Math::PlanePath::ImaginaryHalf -- half-plane replications in three
7       directions
8

SYNOPSIS

10        use Math::PlanePath::ImaginaryBase;
11        my $path = Math::PlanePath::ImaginaryBase->new (radix => 4);
12        my ($x, $y) = $path->n_to_xy (123);
13

DESCRIPTION

15       This is a half-plane variation on the "ImaginaryBase" path.
16
17            54-55 50-51 62-63 58-59 22-23 18-19 30-31 26-27       3
18              \     \     \     \     \     \     \     \
19            52-53 48-49 60-61 56-57 20-21 16-17 28-29 24-25       2
20
21            38-39 34-35 46-47 42-43  6--7  2--3 14-15 10-11       1
22              \     \     \     \     \     \     \     \
23            36-37 32-33 44-45 40-41  4--5  0--1 12-13  8--9   <- Y=0
24
25           -------------------------------------------------
26           -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5
27
28       The pattern can be seen by dividing into blocks,
29
30           +---------------------------------+
31           | 22  23  18  19   30  31  26  27 |
32           |                                 |
33           | 20  21  16  17   28  29  24  25 |
34           +--------+-------+----------------+
35           |  6   7 | 2   3 | 14  15  10  11 |
36           |        +---+---+                |
37           |  4   5 | 0 | 1 | 12  13   8   9 |  <- Y=0
38           +--------+---+---+----------------+
39                      ^
40                     X=0
41
42       N=0 is at the origin, then N=1 replicates it to the right.  Those two
43       repeat above as N=2 and N=3.  Then that 2x2 repeats to the left as N=4
44       to N=7, then 4x2 repeats to the right as N=8 to N=15, and 8x2 above as
45       N=16 to N=31, etc.  The replications are successively to the right,
46       above, left.  The relative layout within a replication is unchanged.
47
48       This is similar to the "ImaginaryBase", but where it repeats in 4
49       directions there's just 3 directions here.  The "ZOrderCurve" is a 2
50       direction replication.
51
52   Radix
53       The "radix" parameter controls the radix used to break N into X,Y.  For
54       example "radix => 4" gives 4x4 blocks, with radix-1 replications of the
55       preceding level at each stage.
56
57            radix => 4
58
59            60 61 62 63 44 45 46 47 28 29 30 31 12 13 14 15      3
60            56 57 58 59 40 41 42 43 24 25 26 27  8  9 10 11      2
61            52 53 54 55 36 37 38 39 20 21 22 23  4  5  6  7      1
62            48 49 50 51 32 33 34 35 16 17 18 19  0  1  2  3  <- Y=0
63
64           --------------------------------------^-----------
65           -12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3
66
67       Notice for X negative the parts replicate successively towards
68       -infinity, so the block N=16 to N=31 is first at X=-4, then N=32 at
69       X=-8, N=48 at X=-12, and N=64 at X=-16 (not shown).
70
71   Digit Order
72       The "digit_order" parameter controls the order digits from N are
73       applied to X and Y.  The default above is "XYX" so the replications go
74       X then Y then negative X.
75
76       "XXY" goes to negative X before Y, so N=2,N=3 goes to negative X before
77       repeating N=4 to N=7 in the Y direction.
78
79           digit_order => "XXY"
80
81           38  39  36  37  46  47  44  45
82           34  35  32  33  42  43  40  41
83            6   7   4   5  14  15  12  13
84            2   3   0   1  10  11   8   9
85           ---------^--------------------
86           -2  -1  X=0  1   2   3   4   5
87
88       The further options are as follows, for six permutations of each 3
89       digits from N.
90
91           digit_order => "YXX"               digit_order => "XnYX"
92           38 39 36 37 46 47 44 45            19 23 18 22 51 55 50 54
93           34 35 32 33 42 43 40 41            17 21 16 20 49 53 48 52
94            6  7  4  5 14 15 12 13             3  7  2  6 35 39 34 38
95            2  3  0  1 10 11  8  9             1  5  0  4 33 37 32 36
96
97           digit_order => "XnXY"              digit_order => "YXnX"
98           37 39 36 38 53 55 52 54            11 15  9 13 43 47 41 45
99           33 35 32 34 49 51 48 50            10 14  8 12 42 46 40 44
100            5  7  4  6 21 23 20 22             3  7  1  5 35 39 33 37
101            1  3  0  2 17 19 16 18             2  6  0  4 34 38 32 36
102
103       "Xn" means the X negative direction.  It's still spaced 2 apart (or
104       whatever radix), so the result is not simply a -X,Y.
105
106   Axis Values
107       N=0,1,4,5,8,9,etc on the X axis (positive and negative) are those
108       integers with a 0 at every third bit starting from the second least
109       significant bit.  This is simply demanding that the bits going to the Y
110       coordinate must be 0.
111
112           X axis Ns = binary ...__0__0__0_     with _ either 0 or 1
113           in octal, digits 0,1,4,5 only
114
115       N=0,1,8,9,etc on the X positive axis have the highest 1-bit in the
116       first slot of a 3-bit group.  Or N=0,4,5,etc on the X negative axis
117       have the high 1 bit in the third slot,
118
119           X pos Ns = binary    1_0__0__0...0__0__0_
120           X neg Ns = binary  10__0__0__0...0__0__0_
121                              ^^^
122                              three bit group
123
124           X pos Ns in octal have high octal digit 1
125           X neg Ns in octal high octal digit 4 or 5
126
127       N=0,2,16,18,etc on the Y axis are conversely those integers with a 0 in
128       two of each three bits, demanding the bits going to the X coordinate
129       must be 0.
130
131           Y axis Ns = binary ..._00_00_00_0    with _ either 0 or 1
132           in octal has digits 0,2 only
133
134       For a radix other than binary the pattern is the same.  Each "_" is any
135       digit of the given radix, and each 0 must be 0.  The high 1 bit for X
136       positive and negative become a high non-zero digit.
137
138   Level Ranges
139       Because the X direction replicates twice for each once in the Y
140       direction the width grows at twice the rate, so after each 3
141       replications
142
143           width = height*height
144
145       For this reason N values for a given Y grow quite rapidly.
146
147   Proth Numbers
148       The Proth numbers, k*2^n+1 for k<2^n, fall in columns on the path.
149
150           *                               *                               *
151
152
153
154           *                               *                               *
155
156
157
158           *                               *                               *
159
160
161
162           *               *               *               *               *
163
164
165
166           *               *               *               *               *
167
168                                   *       *       *       *
169
170           *       *       *       *       *       *       *       *       *
171
172                                   *   *   *   *   *       *
173                                           *
174           *       *       *       *   * *   * *   *       *       *       *
175
176           -----------------------------------------------------------------
177           -31    -23     -15     -7  -3-1 0 3 5   9      17       25     33
178
179       The height of the column is from the zeros in X ending binary
180       ...1000..0001 since this limits the "k" part of the Proth numbers which
181       can have N ending suitably.  Or for X negative ending ...10111...11.
182

FUNCTIONS

184       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
185       classes.
186
187       "$path = Math::PlanePath::ImaginaryBase->new ()"
188       "$path = Math::PlanePath::ImaginaryBase->new (radix => $r, digit_order
189       => $str)"
190           Create and return a new path object.  The choices for "digit_order"
191           are
192
193               "XYX"
194               "XXY"
195               "YXX"
196               "XnYX"
197               "XnXY"
198               "YXnX"
199
200       "($x,$y) = $path->n_to_xy ($n)"
201           Return the X,Y coordinates of point number $n on the path.  Points
202           begin at 0 and if "$n < 0" then the return is an empty list.
203
204       "($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)"
205           The returned range is exact, meaning $n_lo and $n_hi are the
206           smallest and biggest in the rectangle.
207
208   Level Methods
209       "($n_lo, $n_hi) = $path->level_to_n_range($level)"
210           Return "(0, $radix**$level - 1)".
211

SEE ALSO

213       Math::PlanePath, Math::PlanePath::ImaginaryBase,
214       Math::PlanePath::ZOrderCurve
215

HOME PAGE

217       <http://user42.tuxfamily.org/math-planepath/index.html>
218

LICENSE

220       Copyright 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin
221       Ryde
222
223       This file is part of Math-PlanePath.
224
225       Math-PlanePath is free software; you can redistribute it and/or modify
226       it under the terms of the GNU General Public License as published by
227       the Free Software Foundation; either version 3, or (at your option) any
228       later version.
229
230       Math-PlanePath is distributed in the hope that it will be useful, but
231       WITHOUT ANY WARRANTY; without even the implied warranty of
232       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
233       General Public License for more details.
234
235       You should have received a copy of the GNU General Public License along
236       with Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.
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240perl v5.34.0                      2022-01-21 Math::PlanePath::ImaginaryHalf(3)
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