Provided by: libmath-planepath-perl_129-1_all 
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/>.