Math::PlanePath::CincoCurve -- 5x5 self-similar curve
use Math::PlanePath::CincoCurve; my $path = Math::PlanePath::CincoCurve->new; my ($x, $y) = $path->n_to_xy (123);
This is the 5x5 self-similar Cinco curve by John Dennis,
| 4 10--11 14--15--16 35--36 39--40--41 74 71--70 67--66 | | | | | | | | | | | | | 3 9 12--13 18--17 34 37--38 43--42 73--72 69--68 65 | | | | | 2 8 5-- 4 19--20 33 30--29 44--45 52--53--54 63--64 | | | | | | | | | | | 1 7-- 6 3 22--21 32--31 28 47--46 51 56--55 62--61 | | | | | | | Y=0-> 0-- 1-- 2 23--24--25--26--27 48--49--50 57--58--59--60 X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
The base pattern is the N=0 to N=24 part. It repeats transposed and rotated to make the ends join. N=25 to N=49 is a repeat of the base, then N=50 to N=74 is a transpose to go upwards. The sub-part arrangements are as follows.
+------+------+------+------+------+ | 10 | 11 | 14 | 15 | 16 | | | | | | | |----->|----->|----->|----->|----->| +------+------+------+------+------+ |^ 9 | 12 ||^ 13 | 18 ||<-----| || T | T ||| T | T || 17 | || | v|| | v| | +------+------+------+------+------+ |^ 8 | 5 ||^ 4 | 19 || 20 | || T | T ||| T | T || | || | v|| | v|----->| +------+------+------+------+------+ |<-----|<---- |^ 3 | 22 ||<-----| | 7 | 6 || T | T || 21 | | | || | v| | +------+------+------+------+------+ | 0 | 1 |^ 2 | 23 || 24 | | | || T | T || | |----->|----->|| | v|----->| +------+------+------+------+------+
Parts such as 6 going left are the base rotated 180 degrees. The verticals like 2 are a transpose of the base, ie. swap X,Y, and downward vertical like 23 is transpose plus rotate 180 (which is equivalent to a mirror across the anti-diagonal). Notice the base shape fills its sub-part to the left side and the transpose instead fills on the right.
The N values along the X axis are increasing, as are the values along the Y axis. This occurs because the values along the sub-parts of the base are increasing along the X and Y axes, and the other two sides are increasing too when rotated or transposed for sub-parts such as 2 and 23, or 7, 8 and 9.
John Dennis conceived this for use in combination with 2x2 Hilbert and 3x3 meander shapes so that sizes which are products of 2, 3 and 5 can be used for partitioning. Such mixed patterns can't be done with the code here, mainly since a mixture depends on having a target top-level size rather than the unlimited size here.
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::CincoCurve->new ()
Create and return a new path object.
($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.
Math::PlanePath, Math::PlanePath::PeanoCurve, Math::PlanePath::DekkingCurve
John Dennis "Inverse Space-Filling Curve Partitioning of a Global Ocean Model", and source code from COSIM
http://www.cecs.uci.edu/~papers/ipdps07/pdfs/IPDPS-1569010963-paper-2.pdf
http://oceans11.lanl.gov/trac/POP/browser/trunk/pop/source/spacecurve_mod.F90 http://oceans11.lanl.gov/svn/POP/trunk/pop/source/spacecurve_mod.F90
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012, 2013, 2014 Kevin Ryde
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.
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