1Math::PlanePath::PeanoCUusrevre(C3o)ntributed Perl DocumMeanttha:t:iPolnanePath::PeanoCurve(3)
2
6 Math::PlanePath::PeanoCurve -- 3x3 self-similar quadrant traversal
7
9 use Math::PlanePath::PeanoCurve;
10 my $path = Math::PlanePath::PeanoCurve->new;
11 my ($x, $y) = $path->n_to_xy (123);
12 # or another radix digits ...
14 my $path5 = Math::PlanePath::PeanoCurve->new (radix => 5);
15
17 This path is an integer version of the curve described by Peano for
18 filling a unit square,
19 Guiseppe Peano, "Sur Une Courbe, Qui Remplit Toute Une Aire Plane",
21 Mathematische Annalen, volume 36, number 1, 1890, p157-160. DOI
22 10.1007/BF01199438.
23 <http://www.springerlink.com/content/w232301n53960133/>
24 It traverses a quadrant of the plane one step at a time in a self-
26 similar 3x3 pattern,
27 8 60--61--62--63--64--65 78--79--80--...
29 | | |
30 7 59--58--57 68--67--66 77--76--75
31 | | |
32 6 54--55--56 69--70--71--72--73--74
33 |
34 5 53--52--51 38--37--36--35--34--33
35 | | |
36 4 48--49--50 39--40--41 30--31--32
37 | | |
38 3 47--46--45--44--43--42 29--28--27
39 |
40 2 6---7---8---9--10--11 24--25--26
41 | | |
42 1 5---4---3 14--13--12 23--22--21
43 | | |
44 Y=0 0---1---2 15--16--17--18--19--20
45 X=0 1 2 3 4 5 6 7 8 9 ...
47 The start is an S shape of the nine points N=0 to N=8, and then nine of
49 those groups are put together in the same S configuration. The sub-
50 parts are flipped horizontally and/or vertically to make the starts and
51 ends adjacent, so 8 is next to 9, 17 next to 18, etc,
52 60,61,62 --- 63,64,65 78,79,80
54 59,58,57 68,67,55 77,76,75
55 54,55,56 69,70,71 --- 72,73,74
56 |
57 |
58 53,52,51 38,37,36 --- 35,34,33
59 48,49,50 39,40,41 30,31,32
60 47,46,45 --- 44,43,42 29,28,27
61 |
62 |
63 6,7,8 ---- 9,10,11 24,25,26
64 3,4,5 12,13,14 23,22,21
65 0,1,2 15,16,17 --- 18,19,20
66 The process repeats, tripling in size each time.
68 Within a power-of-3 square, 3x3, 9x9, 27x27, 81x81 etc (3^k)x(3^k) at
70 the origin, all the N values 0 to 3^(2*k)-1 are within the square. The
71 top right corner 8, 80, 728, etc is the 3^(2*k)-1 maximum in each.
72 Because each step is by 1, the distance along the curve between two X,Y
74 points is the difference in their N values as given by "xy_to_n()".
75 Radix
77 The "radix" parameter can do the calculation in a base other than 3,
78 using the same kind of direction reversals. For example radix 5 gives
79 5x5 groups,
80 radix => 5
82 4 | 20--21--22--23--24--25--26--27--28--29
84 | | |
85 3 | 19--18--17--16--15 34--33--32--31--30
86 | | |
87 2 | 10--11--12--13--14 35--36--37--38--39
88 | | |
89 1 | 9-- 8-- 7-- 6-- 5 44--43--42--41--40
90 | | |
91 Y=0 | 0-- 1-- 2-- 3-- 4 45--46--47--48--49--50-...
92 |
93 +----------------------------------------------
94 X=0 1 2 3 4 5 6 7 8 9 10
95 If the radix is even then the ends of each group don't join up. For
97 example in radix 4 N=15 isn't next to N=16, nor N=31 to N=32, etc.
98 radix => 4
100 3 | 15--14--13--12 16--17--18--19
102 | | |
103 2 | 8-- 9--10--11 23--22--21--20
104 | | |
105 1 | 7-- 6-- 5-- 4 24--25--26--27
106 | | |
107 Y=0 | 0-- 1-- 2-- 3 31--30--29--28 32--33-...
108 |
109 +------------------------------------------
110 X=0 1 2 4 5 6 7 8 9 10
111 Even sizes can be made to join using other patterns, but this module is
113 just Peano's digit construction. For joining up in 2x2 groupings see
114 "HilbertCurve" (which is essentially the only way to join up in 2x2).
115 For bigger groupings there's various ways.
116 Unit Square
118 Peano's original form was for filling a unit square by mapping a number
119 T in the range 0<T<1 to a pair of X,Y coordinates 0<X<1 and 0<Y<1. The
120 curve is continuous and every such X,Y is reached, so it fills the unit
121 square. A unit cube or higher dimension can be filled similarly by
122 developing three or more coordinates X,Y,Z, etc. Georg Cantor had
123 shown a line is equivalent to the plane, Peano's mapping is a
124 continuous way to do that.
125 The code here could be pressed into service for a fractional T to X,Y
127 by multiplying up by a power of 9 to desired precision then dividing X
128 and Y back by the same power of 3 (perhaps swapping X,Y for which one
129 should be the first ternary digit). Note that if T is a binary
130 floating point then a power of 3 division will round off in general
131 since 1/3 is not exactly representable. (See "HilbertCurve" or
132 "ZOrderCurve" for binary mappings.)
133 Diagonal Lines
135 Moore in
136 E. H. Moore, "On Certain Crinkly Curves", Trans. Am. Math. Soc.,
138 volume 1, number 1, 1900, pages 72-90.
139 <http://www.ams.org/journals/tran/1900-001-01/S0002-9947-1900-1500526-4/>
141 <http://www.ams.org/tran/1900-001-01/S0002-9947-1900-1500526-4/S0002-9947-1900-1500526-4.pdf>
142 <http://www.ams.org/journals/tran/1900-001-04/S0002-9947-1900-1500428-3/>
144 <http://www.ams.org/journals/tran/1900-001-04/S0002-9947-1900-1500428-3/S0002-9947-1900-1500428-3.pdf>
145 draws the curve as a base shape
147 +-----+
149 | |
150 -----+-----+-----
151 | |
152 +-----+
153 with each line segment replaced by the same for the next level (with
155 suitable mirror image in odd segments).
156 The is equivalent to the square form by drawing diagonal lines
158 alternately in the direction of the leading diagonal or opposite
159 diagonal, per the ".." marked lines in the following.
160 +--------+--------+--------+ +--------+--------+--------+
162 | .. | .. | .. | | | | |
163 |6 .. |7 .. |8 .. | | 6--------7--------8 |
164 | .. | .. | .. | | | | | |
165 +--------+--------+--------+ +----|---+--------+--------+
166 | .. | .. | .. | | | | | |
167 | .. 5| .. 4| .. 3| | 5--------4--------3 |
168 | .. | .. | .. | | | | | |
169 +--------+--------+--------+ +--------+--------+----|---+
170 | .. | .. | .. | | | | | |
171 |0 .. |1 .. |2 .. | | 0--------1--------2 |
172 | .. | .. | .. | | | | |
173 +--------+--------+--------+ +--------+--------+--------+
174 X==Y mod 2 "even" points leading-diagonal "/"
176 X!=Y mod 2 "odd" points opposite-diagonal "\"
177 Rounding off the corners of the diagonal form so they don't touch can
179 help show the equivalence,
180 -----7 /
182 / \ /
183 6 -----8
184 |
185 | 4-----
186 \ / \
187 5----- 3
188 |
189 -----1 |
190 / \ /
191 0 -----2
192 Power of 3 Patterns
194 Plotting sequences of values with some connection to ternary digits or
195 powers of 3 will usually give the most interesting patterns on the
196 Peano curve. For example the Mephisto waltz sequence
197 (Math::NumSeq::MephistoWaltz) makes diamond shapes,
198 ** * *** * * *** ** *** ** *** ** ** * *
200 * * ** ** *** ** *** * * ** ** *** ** ***
201 *** ** *** ** ** * *** * *** * * *** **
202 ** *** * * *** * ** ** *** * * *** * **
203 *** ** *** ** ** * *** * *** * * *** **
204 * * ** ** *** ** *** * * ** ** *** ** ***
205 *** ** *** ** ** * *** * *** * * *** **
206 ** *** * * *** * ** ** *** * * *** * **
207 ** * *** * * *** ** *** ** *** ** ** * *
208 * * ** ** *** ** *** * * ** ** *** ** ***
209 ** * *** * * *** ** *** ** *** ** ** * *
210 ** *** * * *** * ** ** *** * * *** * **
211 *** ** *** ** ** * *** * *** * * *** **
212 ** *** * * *** * ** ** *** * * *** * **
213 ** * *** * * *** ** *** ** *** ** ** * *
214 ** *** * * *** * ** ** *** * * *** * **
215 *** ** *** ** ** * *** * *** * * *** **
216 * * ** ** *** ** *** * * ** ** *** ** ***
217 *** ** *** ** ** * *** * *** * * *** **
218 ** *** * * *** * ** ** *** * * *** * **
219 ** * *** * * *** ** *** ** *** ** ** * *
220 * * ** ** *** ** *** * * ** ** *** ** ***
221 ** * *** * * *** ** *** ** *** ** ** * *
222 ** *** * * *** * ** ** *** * * *** * **
223 ** * *** * * *** ** *** ** *** ** ** * *
224 * * ** ** *** ** *** * * ** ** *** ** ***
225 *** ** *** ** ** * *** * *** * * *** **
226 This arises from each 3x3 block in the Mephisto waltz being one of two
228 shapes which are then flipped by the Peano pattern
229 * * _ _ _ *
231 * _ _ or _ * * (inverse)
232 _ _ * * * _
233 0,0,1, 0,0,1, 1,1,0 1,1,0, 1,1,0, 0,0,1
235
237 See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
238 classes.
239 "$path = Math::PlanePath::PeanoCurve->new ()"
241 "$path = Math::PlanePath::PeanoCurve->new (radix => $integer)"
242 Create and return a new path object.
243 The optional "radix" parameter gives the base for digit splitting.
245 The default is ternary "radix => 3".
246 "($x,$y) = $path->n_to_xy ($n)"
248 Return the X,Y coordinates of point number $n on the path. Points
249 begin at 0 and if "$n < 0" then the return is an empty list.
250 Fractional positions give an X,Y position along a straight line
252 between the integer positions. Integer positions are always just 1
253 apart either horizontally or vertically, so the effect is that the
254 fraction part appears either added to or subtracted from X or Y.
255 "$n = $path->xy_to_n ($x,$y)"
257 Return an integer point number for coordinates "$x,$y". Each
258 integer N is considered the centre of a unit square and an "$x,$y"
259 within that square returns N.
260 "($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)"
262 Return a range of N values which occur in a rectangle with corners
263 at $x1,$y1 and $x2,$y2. If the X,Y values are not integers then
264 the curve is treated as unit squares centred on each integer point
265 and squares which are partly covered by the given rectangle are
266 included.
267 The returned range is exact, meaning $n_lo and $n_hi are the
269 smallest and biggest in the rectangle.
270 Level Methods
272 "($n_lo, $n_hi) = $path->level_to_n_range($level)"
273 Return "(0, $radix**(2*$level) - 1)".
274
276 N to X,Y
277 Peano's calculation is based on putting base-3 digits of N alternately
278 to X or Y. From the high end of N a digit is appended to Y then the
279 next appended to X. Beginning at an even digit position in N makes the
280 last digit go to X so the first N=0,1,2 goes along the X axis.
281 At each stage a "complement" state is maintained for X and for Y. When
283 complemented the digit is reversed to 2 - digit, so 0,1,2 becomes
284 2,1,0. This reverses the direction so points like N=12,13,14 shown
285 above go to the left, or groups like 9,10,11 then 12,13,14 then
286 15,16,17 go downwards.
287 The complement is calculated by adding the digits from N which went to
289 the other one of X or Y. So the X complement is the sum of digits
290 which have gone to Y so far. Conversely the Y complement is the sum of
291 digits put to X. If the complement sum is odd then the reversal is
292 done. A bitwise XOR can be used instead of a sum to accumulate
293 odd/even-ness the same way as a sum.
294 When forming the complement state the original digits from N are added,
296 before applying any complementing for putting them to X or Y. If the
297 radix is odd, like the default 3, then complementing doesn't change it
298 mod 2 so either before or after is fine, but if the radix is even then
299 it's not the same.
300 It also works to take the base-3 digits of N from low to high,
302 generating low to high digits in X and Y. When an odd digit is put to
303 X then the low digits of Y so far must be complemented as 22..22 - Y
304 (the 22..22 value being all 2s in base 3, ie. 3^k-1). Conversely if an
305 odd digit is put to Y then X must be complemented. With this approach
306 the high digit position in N doesn't have to be found, but instead peel
307 off digits of N from the low end. But the subtract to complement is
308 then more work if using bignums.
309 X,Y to N
311 The X,Y to N calculation can be done by an inverse of either the high
312 to low or low to high methods above. In both cases digits are put
313 alternately from X and Y onto N, with complement as necessary.
314 For the low to high approach it's not easy to complement just the X
316 digits in the N constructed so far, but it works to build and
317 complement the X and Y digits separately then at the end interleave to
318 make the final N. Complementing is the ternary equivalent of an XOR in
319 binary. On a ternary machine some trit-twiddling could no doubt do it.
320 For the low to high with even radix the complementing is also tricky
322 since changing the accumulated X affects the digits of Y below that,
323 and vice versa. What's the rule? Is it alternate digits which end up
324 complemented? In any case the current "xy_to_n()" code goes high to
325 low which is easier, but means breaking the X,Y inputs into arrays of
326 digits before beginning.
327 N to abs(dX),abs(dY)
329 The curve goes horizontally or vertically according to the number of
330 trailing "2" digits when N is written in ternary,
331 N trailing 2s direction abs(dX) abs(dY)
333 ------------- --------- -------
334 even horizontal 1 0
335 odd vertical 0 1
336 For example N=5 is "12" in ternary has 1 trailing "2" which is odd so
338 the step from N=5 to N=6 is vertical.
339 This works because when stepping from N to N+1 a carry propagates
341 through the trailing 2s to increment the digit above. Digits go
342 alternately to X or Y so odd or even trailing 2s put that carry into an
343 X digit or Y digit.
344 X Y X Y X
346 N ... 2 2 2 2
347 N+1 1 0 0 0 0 carry propagates
348 Rectangle to N Range
350 An easy over-estimate of the maximum N in a region can be had by going
351 to the next bigger (3^k)x(3^k) square enclosing the region. This means
352 the biggest X or Y rounded up to the next power of 3 (perhaps using
353 "log()" if you trust its accuracy), so
354 find k with 3^k > max(X,Y)
356 N_hi = 3^(2k) - 1
357 An exact N range can be found by following the "high to low" N to X,Y
359 procedure above. Start with the easy over-estimate to find a 3^(2k)
360 ternary digit position in N bigger than the desired region, then choose
361 a digit 0,1,2 for X, the biggest which overlaps some of the region. Or
362 if there's an X complement then the smallest digit is the biggest N,
363 again whichever overlaps the region. Then likewise for a digit of Y,
364 etc.
365 Biggest and smallest N must maintain separate complement states as they
367 track down different N digits. A single loop can be used since there's
368 the same "2k" many digits of N to consider for both.
369 The N range of any shape can be done this way, not just a rectangle
371 like "rect_to_n_range()". The procedure only depends on asking whether
372 a one-third sub-part of X or Y overlaps the target region or not.
373
375 This path is in Sloane's Online Encyclopedia of Integer Sequences in
376 several forms,
377 <http://oeis.org/A163528> (etc)
379 A163528 X coordinate
381 A163529 Y coordinate
382 A163530 X+Y coordinate sum
383 A163531 X^2+Y^2 square of distance from origin
384 A163532 dX, change in X -1,0,1
385 A163533 dY, change in Y -1,0,1
386 A014578 abs(dX) from n-1 to n, 1=horiz 0=vertical
387 thue-morse count low 0-bits + 1 mod 2
388 A182581 abs(dY) from n-1 to n, 0=horiz 1=vertical
389 thue-morse count low 0-bits mod 2
390 A163534 direction of each step (up,down,left,right)
391 A163535 direction, transposed X,Y
392 A163536 turn 0=straight,1=right,2=left
393 A163537 turn, transposed X,Y
394 A163342 diagonal sums
395 A163479 diagonal sums divided by 6
396 A163480 N on X axis
398 A163481 N on Y axis
399 A163343 N on X=Y diagonal, 0,4,8,44,40,36,etc
400 A163344 N on X=Y diagonal divided by 4
401 A007417 N+1 of positions of horizontals, ternary even trailing 0s
402 A145204 N+1 of positions of verticals, ternary odd trailing 0s
403 A163332 Peano N -> ZOrder radix=3 N mapping
405 and vice versa since is self-inverse
406 A163333 with ternary digit swaps before and after
407 And taking X,Y points by the Diagonals sequence, then the value of the
409 following sequences is the N of the Peano curve at those positions.
410 A163334 numbering by diagonals, from same axis as first step
412 A163336 numbering by diagonals, from opposite axis
413 A163338 A163334 + 1, Peano starting from N=1
414 A163340 A163336 + 1, Peano starting from N=1
415 "Math::PlanePath::Diagonals" numbers points from the Y axis down, which
417 is the opposite axis to the Peano curve first step along the X axis, so
418 a plain "Diagonals" -> "PeanoCurve" is the "opposite axis" form
419 A163336.
420 These sequences are permutations of the integers since all X,Y
422 positions of the first quadrant are reached eventually. The inverses
423 are as follows. They can be thought of taking X,Y positions in the
424 Peano curve order and then asking what N the Diagonals would put there.
425 A163335 inverse of A163334
427 A163337 inverse of A163336
428 A163339 inverse of A163338
429 A163341 inverse of A163340
430
432 Math::PlanePath, Math::PlanePath::HilbertCurve,
433 Math::PlanePath::ZOrderCurve, Math::PlanePath::AR2W2Curve,
434 Math::PlanePath::BetaOmega, Math::PlanePath::CincoCurve,
435 Math::PlanePath::KochelCurve, Math::PlanePath::WunderlichMeander
436 Math::PlanePath::KochCurve
438
440 <http://user42.tuxfamily.org/math-planepath/index.html>
441
443 Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018 Kevin
444 Ryde
445 This file is part of Math-PlanePath.
447 Math-PlanePath is free software; you can redistribute it and/or modify
449 it under the terms of the GNU General Public License as published by
450 the Free Software Foundation; either version 3, or (at your option) any
451 later version.
452 Math-PlanePath is distributed in the hope that it will be useful, but
454 WITHOUT ANY WARRANTY; without even the implied warranty of
455 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
456 General Public License for more details.
457 You should have received a copy of the GNU General Public License along
459 with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
460perl v5.32.0 2020-07-28 Math::PlanePath::PeanoCurve(3)