Provided by: libmath-planepath-perl_125-1_all 
NAME
Math::PlanePath::ZOrderCurve -- alternate digits to X and Y
SYNOPSIS
use Math::PlanePath::ZOrderCurve;
my $path = Math::PlanePath::ZOrderCurve->new;
my ($x, $y) = $path->n_to_xy (123);
# or another radix digits ...
my $path3 = Math::PlanePath::ZOrderCurve->new (radix => 3);
DESCRIPTION
This path puts points in a self-similar Z pattern described by G.M. Morton,
7 | 42 43 46 47 58 59 62 63
6 | 40 41 44 45 56 57 60 61
5 | 34 35 38 39 50 51 54 55
4 | 32 33 36 37 48 49 52 53
3 | 10 11 14 15 26 27 30 31
2 | 8 9 12 13 24 25 28 29
1 | 2 3 6 7 18 19 22 23
Y=0 | 0 1 4 5 16 17 20 21 64 ...
+---------------------------------------
X=0 1 2 3 4 5 6 7 8
The first four points make a "Z" shape if written with Y going downwards (inverted if drawn upwards as
above),
0---1 Y=0
/
/
2---3 Y=1
Then groups of those are arranged as a further Z, etc, doubling in size each time.
0 1 4 5 Y=0
2 3 --- 6 7 Y=1
/
/
/
8 9 --- 12 13 Y=2
10 11 14 15 Y=3
Within an power of 2 square 2x2, 4x4, 8x8, 16x16 etc (2^k)x(2^k), all the N values 0 to 2^(2*k)-1 are
within the square. The top right corner 3, 15, 63, 255 etc of each is the 2^(2*k)-1 maximum.
Along the X axis N=0,1,4,5,16,17,etc is the integers with only digits 0,1 in base 4. Along the Y axis
N=0,2,8,10,32,etc is the integers with only digits 0,2 in base 4. And along the X=Y diagonal
N=0,3,12,15,etc is digits 0,3 in base 4.
In the base Z pattern it can be seen that transposing to Y,X means swapping parts 1 and 2. This applies
in the sub-parts too so in general if N is at X,Y then changing base 4 digits 1<->2 gives the N at the
transpose Y,X. For example N=22 at X=6,Y=1 is base-4 "112", change 1<->2 is "221" for N=41 at X=1,Y=6.
Power of 2 Values
Plotting N values related to powers of 2 can come out as interesting patterns. For example displaying
the N's which have no digit 3 in their base 4 representation gives
*
* *
* *
* * * *
* *
* * * *
* * * *
* * * * * * * *
* *
* * * *
* * * *
* * * * * * * *
* * * *
* * * * * * * *
* * * * * * * *
* * * * * * * * * * * * * * * *
The 0,1,2 and not 3 makes a little 2x2 "L" at the bottom left, then repeating at 4x4 with again the whole
"3" position undrawn, and so on. This is the Sierpinski triangle (a rotated version of
Math::PlanePath::SierpinskiTriangle). The blanks are also a visual representation of 1-in-4 cross-
products saved by recursive use of the Karatsuba multiplication algorithm.
Plotting the fibbinary numbers (eg. Math::NumSeq::Fibbinary) which are N values with no adjacent 1 bits
in binary makes an attractive tree-like pattern,
*
**
*
****
*
**
* *
********
*
**
*
****
* *
** **
* * * *
****************
* *
** **
* *
**** ****
* *
** **
* * * *
******** ********
* * * *
** ** ** **
* * * *
**** **** **** ****
* * * * * * * *
** ** ** ** ** ** ** **
* * * * * * * * * * * * * * * *
****************************************************************
The horizontals arise from N=...0a0b0c for bits a,b,c so Y=...000 and X=...abc, making those N values
adjacent. Similarly N=...a0b0c0 for a vertical.
Radix
The "radix" parameter can do the same N <-> X/Y digit splitting in a higher base. For example radix 3
makes 3x3 groupings,
radix => 3
5 | 33 34 35 42 43 44
4 | 30 31 32 39 40 41
3 | 27 28 29 36 37 38 45 ...
2 | 6 7 8 15 16 17 24 25 26
1 | 3 4 5 12 13 14 21 22 23
Y=0 | 0 1 2 9 10 11 18 19 20
+--------------------------------------
X=0 1 2 3 4 5 6 7 8
Along the X axis N=0,1,2,9,10,11,etc is integers with only digits 0,1,2 in base 9. Along the Y axis
digits 0,3,6, and along the X=Y diagonal digits 0,4,8. In general for a given radix it's base R*R with
the R many digits of the first RxR block.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$path = Math::PlanePath::ZOrderCurve->new ()"
"$path = Math::PlanePath::ZOrderCurve->new (radix => $r)"
Create and return a new path object. The optional "radix" parameter gives the base for digit
splitting (the default is binary, radix 2).
"($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. The
lines don't overlap, but the lines between bit squares soon become rather long and probably of very
limited use.
"$n = $path->xy_to_n ($x,$y)"
Return an integer point number for coordinates "$x,$y". Each integer N is considered the centre of a
unit square and an "$x,$y" within that square returns N.
"($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)"
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
The coordinate calculation is simple. The bits of X and Y are every second bit of N. So if N = binary
101010 then X=000 and Y=111 in binary, which is the N=42 shown above at X=0,Y=7.
With the "radix" parameter the digits are treated likewise, in the given radix rather than binary.
If N includes a fraction part then it's applied to a straight line towards point N+1. The +1 of N+1
changes X and Y according to how many low radix-1 digits there are in N, and thus in X and Y. In general
if the lowest non radix-1 is in X then
dX=1
dY = - (R^pos - 1) # pos=0 for lowest digit
The simplest case is when the lowest digit of N is not radix-1, so dX=1,dY=0 across.
If the lowest non radix-1 is in Y then
dX = - (R^(pos+1) - 1) # pos=0 for lowest digit
dY = 1
If all digits of X and Y are radix-1 then the implicit 0 above the top of X is considered the lowest non
radix-1 and so the first case applies. In the radix=2 above this happens for instance at N=15 binary
1111 so X = binary 11 and Y = binary 11. The 0 above the top of X is at pos=2 so dX=1, dY=-(2^2-1)=-3.
Rectangle to N Range
Within each row the N values increase as X increases, and within each column N increases with increasing
Y (for all "radix" parameters).
So for a given rectangle the smallest N is at the lower left corner (smallest X and smallest Y), and the
biggest N is at the upper right (biggest X and biggest Y).
OEIS
This path is in Sloane's Online Encyclopedia of Integer Sequences in various forms,
<http://oeis.org/A059905> (etc)
radix=2
A059905 X coordinate
A059906 Y coordinate
A000695 N on X axis (base 4 digits 0,1 only)
A062880 N on Y axis (base 4 digits 0,2 only)
A001196 N on X=Y diagonal (base 4 digits 0,3 only)
A057300 permutation N at transpose Y,X (swap bit pairs)
radix=3
A163325 X coordinate
A163326 Y coordinate
A037314 N on X axis, base 9 digits 0,1,2
A208665 N on X=Y diagonal, base 9 digits 0,3,6
A163327 permutation N at transpose Y,X (swap trit pairs)
radix=4
A126006 permutation N at transpose Y,X (swap digit pairs)
radix=10
A080463 X+Y of radix=10 (from N=1 onwards)
A080464 X*Y of radix=10 (from N=10 onwards)
A080465 abs(X-Y), from N=10 onwards
A051022 N on X axis (base 100 digits 0 to 9)
radix=16
A217558 permutation N at transpose Y,X (swap digit pairs)
And taking X,Y points in the Diagonals sequence then the value of the following sequences is the N of the
"ZOrderCurve" at those positions.
radix=2
A054238 numbering by diagonals, from same axis as first step
A054239 inverse permutation
radix=3
A163328 numbering by diagonals, same axis as first step
A163329 inverse permutation
A163330 numbering by diagonals, opp axis as first step
A163331 inverse permutation
"Math::PlanePath::Diagonals" numbers points from the Y axis down, which is the opposite axis to the
"ZOrderCurve" first step along the X axis, so a transpose is needed to give A054238.
SEE ALSO
Math::PlanePath, Math::PlanePath::PeanoCurve, Math::PlanePath::HilbertCurve,
Math::PlanePath::ImaginaryBase, Math::PlanePath::CornerReplicate, Math::PlanePath::DigitGroups
"http://www.jjj.de/fxt/#fxtbook" (section 1.31.2)
Algorithm::QuadTree, DBIx::SpatialKey
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017 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/>.