/ 
math-image-38
River stage zero No dependents
/ App::MathImage::PlanePath::PeanoCurve

NAME

App::MathImage::PlanePath::PeanoCurve -- self-similar quadrant traversal

SYNOPSIS

use App::MathImage::PlanePath::PeanoCurve;
my $path = App::MathImage::PlanePath::PeanoCurve->new;
my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

This path is an integer version of the curve described by Guiseppe Peano for a unit square. The path traverses a quadrant of the plane one step at a time in a self-similar 3x3 pattern,

y=8   60--61--62--63--64--65  78--79--80--...
       |                   |   |
y=7   59--58--57  68--67--66  77--76--75
               |   |                   |
y=6   54--55--56  69--70--71--72--73--74
       |
y=5   53--52--51  38--37--36--35--34--33
               |   |                   |
y=4   48--49--50  39--40--41  30--31--32
       |                   |   |
y=3   47--46--45--44--43--42  29--28--27
                                       |
y=2    6---7---8---9--10--11  24--25--26
       |                   |   |
y=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 points 0 to 8, and then nine of those are put together in the same configuration, with sub-parts flipped horizontally and/or vertically to make the starts and ends adjacent (8 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 each time.

Within a power-of-3 square 3x3, 9x9, 27x27, 81x81 etc (3^k)x(3^k), 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.

Unit Square

Peano's form is based on 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 X,Y is reached so it fills the unit square. A unit cube can be filled by developing three coordinates X,Y,Z similarly. (Georg Cantor had shown a line is equivalent to a surface, Peano's mapping is a continuous mapping doing that.)

The code here can 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,Y back by the same power of 3, perhaps transposing X,Y for which one you want to have the first digit. If T is floating point then a power of 3 division will be rounded off since a division by 3 is not exactly representable in binary, in general. (See HilbertCurve or ZOrderCurve for binary based paths.)

OEIS

This path is in Sloane's OEIS in several forms, eg.

http://www.oeis.org/A163528

A163528    X coordinate
A163529    Y coordinate
A163530    coordinate sum X+Y
A163531    square of distance from origin (X^2+Y^2)
A163532    X change -1,0,1
A163533    Y change -1,0,1
A163534    absolute direction of each step (up,down,left,right)
A163535    absolute direction, transpose X,Y
A163536    relative direction (ahead,left,right)
A163537    relative direction, transpose X,Y
A163342    diagonals summed
A163343    central diagonal  0,4,8,44,40,36,etc
A163344    central diagonal divided by 4
A163480    row at X=0
A163481    column at Y=0

And taking squares of the plane in diagonals sequence, each value 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    one-based, ie. A163334 + 1

Math::PlanePath::Diagonals numbers from the Y axis down, which is the opposite axis to the Peano curve first step along the X axis, which means Diagonals+PeanoCurve is the "opposite axis" A163336 sequence.

The sequences are in each case permutations of the integers since all X,Y positions 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

FORMULAS

Peano's calculation is based on splitting the base-3 digits of N alternately between X and Y. Starting from the high end of N a digit is appended to X then the next appended to Y, arranging for the last to go to X. At each stage a "complement" state is maintained for X and Y. When complemented the digit is reversed to 2 - digit, so 0,1,2 becomes 2,1,0. The represents the reverse for points like N=12,13,14 shown above.

The complement is calculated by adding up the N digits which went to the other of X or Y. So the X complement is the sum of digits which have been appended to Y so far, and conversely Y complement is the sum of digits applied to X. If the sum is odd then the reversal is done. The odd/even is not changed by the reversal itself, so it doesn't matter if the digit is taken before or after. An XOR can be used instead of a sum, that too maintaining the desired odd/even.

It also works to take the base-3 digits of N from low to high, applying digits to the high end of X and Y successively. When an odd digit, ie. a 1, is put onto X then the digits of Y so far must be complemented as 22..22 - Y, the 22..22 value being all 2s in base 3. Conversely if a digit 1 is added to Y then X must be complemented. With this approach the high digits of N don't have to be found, instead just peeled off the low end, but the full subtract for the complement would be more work if using bignums.

The X,Y to N calculation can be done by an inverse of either method, putting digits alternately from X and Y onto N, with complement as necessary. For the low to high approach complementing just the X digits in the constructed N isn't easy. The X and Y digits to go into N can be built up separately then interleaved to make the final N. The complementing is the equivalent of an XOR in binary. On a ternary machine some trit-twidding could no doubt do it.

In the current code n_to_xy and xy_to_n both go low to high as that seems a bit easier than finding the high ternary digits of the inputs.

FUNCTIONS

$path = App::MathImage::PlanePath::PeanoCurve->new ()

Create and return a new path object.

($x,$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 positions 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 an integer point number for coordinates $x,$y. Each integer N is considered the centre of a unit square an $x,$y within that square returns N.

SEE ALSO

Math::PlanePath, Math::PlanePath::HilbertCurve Math::PlanePath::ZOrderCurve

Guiseppe Peano, "Sur une courbe, qui remplit toute une aire plane", Mathematische Annalen, volume 36, number 1, 1890, p157-160

DOI 10.1007/BF01199438
http://www.springerlink.com/content/w232301n53960133/

HOME PAGE

http://user42.tuxfamily.org/math-image/index.html

LICENSE

Math-Image is Copyright 2010, 2011 Kevin Ryde

Math-Image 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-Image 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-Image. If not, see <http://www.gnu.org/licenses/>.