oracular (3) Math::PlanePath::PeanoCurve.3pm.gz
NAME
Math::PlanePath::PeanoCurve -- 3x3 self-similar quadrant traversal
SYNOPSIS
use Math::PlanePath::PeanoCurve;
my $path = Math::PlanePath::PeanoCurve->new;
my ($x, $y) = $path->n_to_xy (123);
# or another radix digits ...
my $path5 = Math::PlanePath::PeanoCurve->new (radix => 5);
DESCRIPTION
This path is an integer version of the curve described by Peano for filling a unit square,
Giuseppe Peano, "Sur Une Courbe, Qui Remplit Toute Une Aire Plane", Mathematische Annalen, volume 36,
number 1, 1890, pages 157-160. DOI 10.1007/BF01199438. <https://eudml.org/doc/157489>,
<https://link.springer.com/article/10.1007/BF01199438>
It traverses a quadrant of the plane one step at a time in a self-similar 3x3 pattern,
8 60--61--62--63--64--65 78--79--80--...
| | |
7 59--58--57 68--67--66 77--76--75
| | |
6 54--55--56 69--70--71--72--73--74
|
5 53--52--51 38--37--36--35--34--33
| | |
4 48--49--50 39--40--41 30--31--32
| | |
3 47--46--45--44--43--42 29--28--27
|
2 6---7---8---9--10--11 24--25--26
| | |
1 5---4---3 14--13--12 23--22--21
| | |
Y=0 0---1---2 15--16--17--18--19--20
X=0 1 2 3 4 5 6 7 8 9 ...
The start is an S shape of the nine points N=0 to N=8, and then nine of those groups are put together in
the same S configuration. The sub-parts are flipped horizontally and/or vertically to make the starts
and ends adjacent, so 8 is next to 9, 17 next to 18, etc,
60,61,62 --- 63,64,65 78,79,80
59,58,57 68,67,55 77,76,75
54,55,56 69,70,71 --- 72,73,74
|
|
53,52,51 38,37,36 --- 35,34,33
48,49,50 39,40,41 30,31,32
47,46,45 --- 44,43,42 29,28,27
|
|
6,7,8 ---- 9,10,11 24,25,26
3,4,5 12,13,14 23,22,21
0,1,2 15,16,17 --- 18,19,20
The process repeats, tripling in size each time.
Within a power-of-3 square, 3x3, 9x9, 27x27, 81x81 etc (3^k)x(3^k) at the origin, all the N values 0 to
3^(2*k)-1 are within the square. The top right corner 8, 80, 728, etc is the 3^(2*k)-1 maximum in each.
Because each step is by 1, the distance along the curve between two X,Y points is the difference in their
N values as given by "xy_to_n()".
Radix
The "radix" parameter can do the calculation in a base other than 3, using the same kind of direction
reversals. For example radix 5 gives 5x5 groups,
radix => 5
4 | 20--21--22--23--24--25--26--27--28--29
| | |
3 | 19--18--17--16--15 34--33--32--31--30
| | |
2 | 10--11--12--13--14 35--36--37--38--39
| | |
1 | 9-- 8-- 7-- 6-- 5 44--43--42--41--40
| | |
Y=0 | 0-- 1-- 2-- 3-- 4 45--46--47--48--49--50-...
|
+----------------------------------------------
X=0 1 2 3 4 5 6 7 8 9 10
If the radix is even then the ends of each group don't join up. For example in radix 4 N=15 isn't next
to N=16, nor N=31 to N=32, etc.
radix => 4
3 | 15--14--13--12 16--17--18--19
| | |
2 | 8-- 9--10--11 23--22--21--20
| | |
1 | 7-- 6-- 5-- 4 24--25--26--27
| | |
Y=0 | 0-- 1-- 2-- 3 31--30--29--28 32--33-...
|
+------------------------------------------
X=0 1 2 4 5 6 7 8 9 10
Even sizes can be made to join using other patterns, but this module is just Peano's digit construction.
For joining up in 2x2 groupings see "HilbertCurve" (which is essentially the only way to join up in 2x2).
For bigger groupings there's various ways.
Unit Square
Peano's original form was for filling a unit square by mapping a number T in the range 0<T<1 to a pair of
X,Y coordinates 0<X<1 and 0<Y<1. The curve is continuous and every such X,Y is reached by some T, so it
fills the unit square. A unit cube or higher dimension can be filled similarly by developing three or
more coordinates X,Y,Z, etc. Cantor had shown a line is equivalent to the plane, Peano's mapping is a
continuous way to do that.
The code here could be pressed into service for a fractional T to X,Y by multiplying up by a power of 9
to desired precision then dividing X and Y back by the same power of 3 (perhaps swapping X,Y for which
one should be the first ternary digit). Note that if T is a binary floating point then a power of 3
division will round off in general since 1/3 is not exactly representable. (See "HilbertCurve" or
"ZOrderCurve" for binary mappings.)
Sometimes the curve is drawn with line segments crossing unit squares. See PeanoDiagonals for that sort
of path.
Power of 3 Patterns
Plotting sequences of values with some connection to ternary digits or powers of 3 will usually give the
most interesting patterns on the Peano curve. For example the Mephisto waltz sequence
(Math::NumSeq::MephistoWaltz) makes diamond shapes,
** * *** * * *** ** *** ** *** ** ** * *
* * ** ** *** ** *** * * ** ** *** ** ***
*** ** *** ** ** * *** * *** * * *** **
** *** * * *** * ** ** *** * * *** * **
*** ** *** ** ** * *** * *** * * *** **
* * ** ** *** ** *** * * ** ** *** ** ***
*** ** *** ** ** * *** * *** * * *** **
** *** * * *** * ** ** *** * * *** * **
** * *** * * *** ** *** ** *** ** ** * *
* * ** ** *** ** *** * * ** ** *** ** ***
** * *** * * *** ** *** ** *** ** ** * *
** *** * * *** * ** ** *** * * *** * **
*** ** *** ** ** * *** * *** * * *** **
** *** * * *** * ** ** *** * * *** * **
** * *** * * *** ** *** ** *** ** ** * *
** *** * * *** * ** ** *** * * *** * **
*** ** *** ** ** * *** * *** * * *** **
* * ** ** *** ** *** * * ** ** *** ** ***
*** ** *** ** ** * *** * *** * * *** **
** *** * * *** * ** ** *** * * *** * **
** * *** * * *** ** *** ** *** ** ** * *
* * ** ** *** ** *** * * ** ** *** ** ***
** * *** * * *** ** *** ** *** ** ** * *
** *** * * *** * ** ** *** * * *** * **
** * *** * * *** ** *** ** *** ** ** * *
* * ** ** *** ** *** * * ** ** *** ** ***
*** ** *** ** ** * *** * *** * * *** **
This arises from each 3x3 block in the Mephisto waltz being one of two shapes which are then flipped by
the Peano pattern
* * _ _ _ *
* _ _ or _ * * (inverse)
_ _ * * * _
0,0,1, 0,0,1, 1,1,0 1,1,0, 1,1,0, 0,0,1
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$path = Math::PlanePath::PeanoCurve->new ()"
"$path = Math::PlanePath::PeanoCurve->new (radix => $integer)"
Create and return a new path object.
The optional "radix" parameter gives the base for digit splitting. The default is ternary "radix =>
3".
"($x,$y) = $path->n_to_xy ($n)"
Return the X,Y coordinates of point number $n on the path. Points begin at 0 and if "$n < 0" then
the return is an empty list.
Fractional $n give an X,Y position along a straight line between the integer positions. Integer
positions are always just 1 apart either horizontally or vertically, so the effect is that the
fraction part appears either added to or subtracted from X or Y.
"$n = $path->xy_to_n ($x,$y)"
Return the integer point number for coordinates "$x,$y". Each integer N is considered the centre of
a unit square and an "$x,$y" within that square returns N.
"($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)"
Return the range of N values which occur the a rectangle with corners at $x1,$y1 and $x2,$y2. If the
X,Y values are not integers then the curve is treated as unit squares centred on each integer point
and squares which are partly covered by the given rectangle are included.
The returned range is exact, meaning $n_lo and $n_hi are the smallest and biggest in the rectangle.
Level Methods
"($n_lo, $n_hi) = $path->level_to_n_range($level)"
Return "(0, $radix**(2*$level) - 1)".
FORMULAS
N to X,Y
Peano's calculation is based on putting base-3 digits of N alternately to X or Y. From the high end of
N, a digit goes to Y then the next goes to X. Beginning at an even digit position in N makes the last
digit go to X so the first N=0,1,2 is along the X axis.
At each stage a "complement" state is maintained for X and for Y. When complemented, the digit is
reversed to 2 - digit, so 0,1,2 becomes 2,1,0. This reverses the direction so points like N=12,13,14
shown above go leftwards, or groups like 9,10,11 then 12,13,14 then 15,16,17 go downwards.
The complement is calculated by adding the digits from N which went to the other one of X or Y. So the X
complement is the sum of digits which have gone to Y so far. Conversely the Y complement is the sum of
digits put to X. If the complement sum is odd then the reversal is done. A bitwise XOR can be used
instead of a sum to accumulate odd/even-ness the same way as a sum.
When forming the complement state, the original digits from N are added, before applying any
complementing for putting them to X or Y. If the radix is odd, like the default 3, then complementing
doesn't change it mod 2 so before or after are the same, but if the radix is even then it's not the same.
It also works to take the base-3 digits of N from low to high, generating low to high digits in X and Y.
If an odd digit is put to X then the low digits of Y so far must be complemented as 22..22 - Y (the
22..22 value being all 2s in base 3, ie. 3^k-1). Conversely if an odd digit is put to Y then X must be
complemented. With this approach, the high digit position in N doesn't have to be found, just peel off
digits of N from the low end. But the subtract to complement is then more work if using bignums.
X,Y to N
The X,Y to N calculation can be done by an inverse of either the high to low or low to high methods
above. In both cases digits are put alternately from X and Y into N, with complement as necessary.
For the low to high approach, it's not easy to complement just the X digits in the N constructed so far,
but it works to build and complement the X and Y digits separately then at the end interleave to make the
final N. Complementing is the ternary equivalent of an XOR in binary. On a ternary machine maybe some
trit-twiddling would do it.
For low to high with even radix, the complementing is also tricky since changing the accumulated X
affects the digits of Y below that, and vice versa. What's the rule? Is it alternate digits which end
up complemented? In any case the current "xy_to_n()" code goes high to low which is easier, but means
breaking the X,Y inputs into arrays of digits before beginning.
N on Axes
N on the X axis is all Y digits 0 in the X,Y to N described above. This means N is the digits of X, and
then digit 0 or 2 at each Y position according to odd or even sum of X digits above. The Y digits are at
odd positions so the 0 or 2 ternary is 0 or 6 for N in base-9.
N on X axis = 0,1,2, 15,16,17, 18,19,20, 141, ... (A163480)
ternary 0,1,2, 120,121,122, 200,201,202, 12020
The Y axis is similar but the X digits are at even positions.
N on Y axis = 0,5,6, 47,48,53, 54,59,60, 425, ... (A163481)
ternary 0,12,20, 1202,1210,1222, 2000,2012,2020, 120202
N on the X=Y diagonal has the ternary digits of position d go to both X and Y and so both complemented
according to sum of digits of d above. That transformation within d is the ternary reflected Gray code.
Gray3(d) = ternary flip 0<->2 when sum of digits above is odd
= 0,1,2, 5,4,3, 6,7,8, 17, ... (A128173)
ternary 0,1,2, 12,11,10, 20,21,22, 122, ...
N on X=Y diag = ternary Gray3(d) and 0,1,2 -> 0,4,8 base9,
which is 4*digit
= 0,4,8, 44,40,36, 72,76,80, 404, ... (A163343)
ternary 0,11,22, 1122,1111,1100, 2200,2211,2222, 112222,
N to abs(dX),abs(dY)
The curve goes horizontally or vertically according to the number of trailing "2" digits when N is
written in ternary,
N trailing 2s direction abs(dX) abs(dY)
------------- --------- ------- -------
even horizontal 1 0
odd vertical 0 1
abs(dX) = 1,1,0, 1,1,0, 1,1,1, 1,1,0, 1,1,0, 1,1,1, ... (A014578)
abs(dY) = 0,0,1, 0,0,1, 0,0,0, 0,0,1, 0,0,1, 0,0,0, ... (A182581)
For example N=5 is "12" in ternary has 1 trailing "2" which is odd so the step from N=5 to N=6 is
vertical.
This works because when stepping from N to N+1 a carry propagates through the trailing 2s to increment
the digit above. Digits go alternately to X or Y so odd or even trailing 2s put that carry into an X
digit or Y digit.
X Y X Y X
N ... 2 2 2 2
N+1 1 0 0 0 0 carry propagates
Rectangle to N Range
An easy over-estimate of the maximum N in a region can be had by going to the next bigger (3^k)x(3^k)
square enclosing the region. This means the biggest X or Y rounded up to the next power of 3 (perhaps
using "log()" if you trust its accuracy), so
find k with 3^k > max(X,Y)
N_hi = 3^(2k) - 1
An exact N range can be found by following the "high to low" N to X,Y procedure above. Start with the
easy over-estimate to find a 3^(2k) ternary digit position in N bigger than the desired region, then
choose a digit 0,1,2 for X, the biggest which overlaps some of the region. Or if there's an X complement
then the smallest digit is the biggest N, again whichever overlaps the region. Then likewise for a digit
of Y, etc.
Biggest and smallest N must maintain separate complement states as they track down different N digits. A
single loop can be used since there's the same "2k" many digits of N to consider for both.
The N range of any shape can be done this way, not just a rectangle like "rect_to_n_range()". The
procedure only depends on asking whether a one-third sub-part of X or Y overlaps the target region or
not.
OEIS
This path is in Sloane's Online Encyclopedia of Integer Sequences in several forms,
<http://oeis.org/A163528> (etc)
A163528 X coordinate
A163529 Y coordinate
A163530 X+Y coordinate sum
A163531 X^2+Y^2 square of distance from origin
A163532 dX, change in X -1,0,1
A163533 dY, change in Y -1,0,1
A014578 abs(dX) from n-1 to n, 1=horiz 0=vertical
A182581 abs(dY) from n-1 to n, 0=horiz 1=vertical
A163534 direction mod 4 of each step (ENWS)
A163535 direction mod 4, transposed X,Y
A163536 turn 0=straight,1=right,2=left
A163537 turn, transposed X,Y
A163342 diagonal sums
A163479 diagonal sums divided by 6
A163480 N on X axis
A163481 N on Y axis
A163343 N on X=Y diagonal, 0,4,8,44,40,36,etc
A163344 N on X=Y diagonal divided by 4
A007417 N+1 of positions of horizontals, ternary even trailing 0s
A145204 N+1 of positions of verticals, ternary odd trailing 0s
A163332 Peano N <-> ZOrder radix=3 N mapping (self-inverse)
A163333 with ternary digit swaps before and after
And taking X,Y points by the Diagonals sequence, then the value of the following sequences is the N of
the Peano curve at those positions.
A163334 numbering by diagonals, from same axis as first step
A163336 numbering by diagonals, from opposite axis
A163338 A163334 + 1, Peano starting from N=1
A163340 A163336 + 1, Peano starting from N=1
"Math::PlanePath::Diagonals" numbers points from the Y axis down, which is the opposite axis to the Peano
curve first step along the X axis, so a plain "Diagonals" -> "PeanoCurve" is the "opposite axis" form
A163336.
These sequences are permutations of the integers since all X,Y positions of the first quadrant are
reached eventually. The inverses are as follows. They can be thought of taking X,Y positions in the
Peano curve order and then asking what N the Diagonals would put there.
A163335 inverse of A163334
A163337 inverse of A163336
A163339 inverse of A163338
A163341 inverse of A163340
SEE ALSO
Math::PlanePath, Math::PlanePath::PeanoDiagonals, Math::PlanePath::HilbertCurve,
Math::PlanePath::ZOrderCurve, Math::PlanePath::AR2W2Curve, Math::PlanePath::BetaOmega,
Math::PlanePath::CincoCurve, Math::PlanePath::KochelCurve, Math::PlanePath::WunderlichMeander
Math::PlanePath::KochCurve
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde
This file is part of Math-PlanePath.
Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the GNU
General Public License as published by the Free Software Foundation; either version 3, or (at your
option) any later version.
Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even
the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
License for more details.
You should have received a copy of the GNU General Public License along with Math-PlanePath. If not, see
<http://www.gnu.org/licenses/>.