NAME
Math::PlanePath::ZOrderCurve  2x2 selfsimilar Z shape digits
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 selfsimilar 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
The first four points make a "Z" shape if written with Y going downwards (inverted if drawn upwards as above),
01 Y=0
/
/
23 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.
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 1in4 crossproducts 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 treelike 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 sort of N > X/Y digit splitting in a higher base. For example radix 3 makes 3x3 groupings,
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
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for the 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.
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.
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).
SEE ALSO
Math::PlanePath, Math::PlanePath::PeanoCurve, Math::PlanePath::HilbertCurve, Math::PlanePath::ImaginaryBase
http://www.jjj.de/fxt/#fxtbook
(section 1.31.2)
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
Copyright 2010, 2011 Kevin Ryde
This file is part of MathPlanePath.
MathPlanePath 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.
MathPlanePath 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 MathPlanePath. If not, see <http://www.gnu.org/licenses/>.