1Math::PlanePath::PowerAUrsreary(C3o)ntributed Perl DocumMeanttha:t:iPolnanePath::PowerArray(3)
2
6 Math::PlanePath::PowerArray -- array by powers
7
9 use Math::PlanePath::PowerArray;
10 my $path = Math::PlanePath::PowerArray->new (radix => 2);
11 my ($x, $y) = $path->n_to_xy (123);
12
14 This is a split of N into an odd part and power of 2,
15 14 | 29 58 116 232 464 928 1856 3712 7424 14848
17 13 | 27 54 108 216 432 864 1728 3456 6912 13824
18 12 | 25 50 100 200 400 800 1600 3200 6400 12800
19 11 | 23 46 92 184 368 736 1472 2944 5888 11776
20 10 | 21 42 84 168 336 672 1344 2688 5376 10752
21 9 | 19 38 76 152 304 608 1216 2432 4864 9728
22 8 | 17 34 68 136 272 544 1088 2176 4352 8704
23 7 | 15 30 60 120 240 480 960 1920 3840 7680
24 6 | 13 26 52 104 208 416 832 1664 3328 6656
25 5 | 11 22 44 88 176 352 704 1408 2816 5632
26 4 | 9 18 36 72 144 288 576 1152 2304 4608
27 3 | 7 14 28 56 112 224 448 896 1792 3584
28 2 | 5 10 20 40 80 160 320 640 1280 2560
29 1 | 3 6 12 24 48 96 192 384 768 1536
30 Y=0 | 1 2 4 8 16 32 64 128 256 512
31 +-----------------------------------------------------------
32 X=0 1 2 3 4 5 6 7 8 9
33 For N=odd*2^k the coordinates are X=k, Y=(odd-1)/2. The X coordinate
35 is how many factors of 2 can be divided out. The Y coordinate counts
36 odd integers 1,3,5,7,etc as 0,1,2,3,etc. This is clearer by writing N
37 values in binary,
38 N values in binary
40 6 | 1101 11010 110100 1101000 11010000 110100000
42 5 | 1011 10110 101100 1011000 10110000 101100000
43 4 | 1001 10010 100100 1001000 10010000 100100000
44 3 | 111 1110 11100 111000 1110000 11100000
45 2 | 101 1010 10100 101000 1010000 10100000
46 1 | 11 110 1100 11000 110000 1100000
47 Y=0 | 1 10 100 1000 10000 100000
48 +----------------------------------------------------------
49 X=0 1 2 3 4 5
50 Radix
52 The "radix" parameter can do the same dividing out in a higher base.
53 For example radix 3 divides out factors of 3,
54 radix => 3
56 9 | 14 42 126 378 1134 3402 10206 30618
58 8 | 13 39 117 351 1053 3159 9477 28431
59 7 | 11 33 99 297 891 2673 8019 24057
60 6 | 10 30 90 270 810 2430 7290 21870
61 5 | 8 24 72 216 648 1944 5832 17496
62 4 | 7 21 63 189 567 1701 5103 15309
63 3 | 5 15 45 135 405 1215 3645 10935
64 2 | 4 12 36 108 324 972 2916 8748
65 1 | 2 6 18 54 162 486 1458 4374
66 Y=0 | 1 3 9 27 81 243 729 2187
67 +------------------------------------------------
68 X=0 1 2 3 4 5 6 7
69 N=1,3,9,27,etc on the X axis is the powers of 3.
71 N=1,2,4,5,7,etc on the Y axis is the integers N=1or2 mod 3, ie. those
73 not a multiple of 3. Notice if Y=1or2 mod 4 then the N values in that
74 row are all even, or if Y=0or3 mod 4 then the N values are all odd.
75 radix => 3, N values in ternary
77 6 | 101 1010 10100 101000 1010000 10100000
79 5 | 22 220 2200 22000 220000 2200000
80 4 | 21 210 2100 21000 210000 2100000
81 3 | 12 120 1200 12000 120000 1200000
82 2 | 11 110 1100 11000 110000 1100000
83 1 | 2 20 200 2000 20000 200000
84 Y=0 | 1 10 100 1000 10000 100000
85 +----------------------------------------------------
86 X=0 1 2 3 4 5
87 Boundary Length
89 The points N=1 to N=2^k-1 inclusive have a boundary length
90 boundary = 2^k + 2k
92 For example N=1 to N=7 is
94 +---+
96 | 7 |
97 + +
98 | 5 |
99 + +---+
100 | 3 6 |
101 + +---+
102 | 1 2 4 |
103 +---+---+---+
104 The height is the odd numbers, so 2^(k-1). The width is the power k.
106 So total boundary 2*height+2*width = 2^k + 2k.
107 If N=2^k is included then it's on the X axis and so add 2, for boundary
109 = 2^k + 2k + 2.
110 For other radix the calculation is similar
112 boundary = 2 * (radix-1) * radix^(k-1) + 2*k
114 For example radix=3, N=1 to N=8 is
116 8
118 7
119 5
120 4
121 2 6
122 1 3
123 The height is the non-multiples of the radix, so (radix-1)/radix *
125 radix^k. The width is the power k again. So total boundary =
126 2*height+2*width.
127
129 See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path
130 classes.
131 "$path = Math::PlanePath::PowerArray->new ()"
133 Create and return a new path object.
134 "($x,$y) = $path->n_to_xy ($n)"
136 Return the X,Y coordinates of point number $n on the path. Points
137 begin at 1 and if "$n < 0" then the return is an empty list.
138 "$n = $path->xy_to_n ($x,$y)"
140 Return the N point number at coordinates "$x,$y". If "$x<0" or
141 "$y<0" then there's no N and the return is "undef".
142 N values grow rapidly with $x. Pass in a number type such as
144 "Math::BigInt" to preserve precision.
145 "($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)"
147 The returned range is exact, meaning $n_lo and $n_hi are the
148 smallest and biggest in the rectangle.
149
151 Rectangle to N Range
152 Within each row increasing X is increasing N, and in each column
153 increasing Y is increasing N. So in a rectangle the lower left corner
154 is the minimum N and the upper right is the maximum N.
155 | N max
157 | ----------+
158 | | ^ |
159 | | | |
160 | | ----> |
161 | +----------
162 | N min
163 +-------------------
164 N to Turn Left or Right
166 The turn left or right is given by
167 radix = 2 left at N==0 mod radix and N==1mod4, right otherwise
169 radix >= 3 left at N==0 mod radix
171 right at N=1 or radix-1 mod radix
172 straight otherwise
173 The points N!=0 mod radix are on the Y axis and those N==0 mod radix
175 are off the axis. For that reason the turn at N==0 mod radix is to the
176 left,
177 |
179 C--
180 ---
181 A--__ -- point B is N=0 mod radix,
182 | --- B turn left A-B-C is left
183 For radix>=3 the turns at A and C are to the right, since the point
185 before A and after C is also on the Y axis. For radix>=4 there's of
186 run of points on the Y axis which are straight.
187 For radix=2 the "B" case N=0 mod 2 applies, but for the A,C points in
189 between the turn alternates left or right.
190 1-- N=1 mod 4 3-- N=3 mod 4
192 \ -- turn left \ -- turn right
193 \ -- \ --
194 2 -- 2 --
195 -- --
196 -- --
197 0 4
198 Points N=2 mod 4 are X=1 and Y=N/2 whereas N=0 mod 4 has 2 or more
200 trailing 0 bits so X>1 and Y<N/2.
201 N mod 4 turn
203 ------- ------
204 0 left for radix=2
205 1 left
206 2 left
207 3 right
208
210 Entries in Sloane's Online Encyclopedia of Integer Sequences related to
211 this path include
212 <http://oeis.org/A007814> (etc)
214 radix=2
216 A007814 X coordinate, count low 0-bits of N
217 A006519 2^X
218 A025480 Y coordinate of N-1, ie. seq starts from N=0
220 A003602 Y+1, being k for which N=(2k-1)*2^m
221 A153733 2*Y of N-1, strip low 1 bits
222 A000265 2*Y+1, strip low 0 bits
223 A094267 dX, change count low 0-bits
225 A050603 abs(dX)
226 A108715 dY, change in Y coordinate
227 A000079 N on X axis, powers 2^X
229 A057716 N not on X axis, the non-powers-of-2
230 A005408 N on Y axis (X=0), the odd numbers
232 A003159 N in X=even columns, even trailing 0 bits
233 A036554 N in X=odd columns
234 A014480 N on X=Y diagonal, (2n+1)*2^n
236 A118417 N on X=Y+1 diagonal, (2n-1)*2^n
237 (just below X=Y diagonal)
238 A054582 permutation N by diagonals, upwards
240 A135764 permutation N by diagonals, downwards
241 A075300 permutation N-1 by diagonals, upwards
242 A117303 permutation N at transpose X,Y
243 A100314 boundary length for N=1 to N=2^k-1 inclusive
245 being 2^k+2k
246 A131831 same, after initial 1
247 A052968 half boundary length N=1 to N=2^k inclusive
248 being 2^(k-1)+k+1
249 radix=3
251 A007949 X coordinate, power-of-3 dividing N
252 A000244 N on X axis, powers 3^X
253 A001651 N on Y axis (X=0), not divisible by 3
254 A007417 N in X=even columns, even trailing 0 digits
255 A145204 N in X=odd columns (extra initial 0)
256 A141396 permutation, N by diagonals down from Y axis
257 A191449 permutation, N by diagonals up from X axis
258 A135765 odd N by diagonals, deletes the Y=1,2mod4 rows
259 A000975 Y at N=2^k, being binary "10101..101"
260 radix=4
262 A000302 N on X axis, powers 4^X
263 radix=5
265 A112765 X coordinate, power-of-5 dividing N
266 A000351 N on X axis, powers 5^X
267 radix=6
269 A122841 X coordinate, power-of-6 dividing N
270 radix=10
272 A011557 N on X axis, powers 10^X
273 A067251 N on Y axis, not a multiple of 10
274 A151754 Y coordinate of N=2^k, being floor(2^k*9/10)
275
277 Math::PlanePath, Math::PlanePath::WythoffArray,
278 Math::PlanePath::ZOrderCurve
279 David M. Bradley "Counting Ordered Pairs", Mathematics Magazine, volume
281 83, number 4, October 2010, page 302, DOI 10.4169/002557010X528032.
282 <http://www.math.umaine.edu/~bradley/papers/JStor002557010X528032.pdf>
283
285 <http://user42.tuxfamily.org/math-planepath/index.html>
286
288 Copyright 2012, 2013, 2014, 2015, 2016, 2017, 2018 Kevin Ryde
289 This file is part of Math-PlanePath.
291 Math-PlanePath is free software; you can redistribute it and/or modify
293 it under the terms of the GNU General Public License as published by
294 the Free Software Foundation; either version 3, or (at your option) any
295 later version.
296 Math-PlanePath is distributed in the hope that it will be useful, but
298 WITHOUT ANY WARRANTY; without even the implied warranty of
299 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
300 General Public License for more details.
301 You should have received a copy of the GNU General Public License along
303 with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
304perl v5.32.0 2020-07-28 Math::PlanePath::PowerArray(3)