Math::PlanePath::CornerReplicate -- replicating U parts
use Math::PlanePath::CornerReplicate; my $path = Math::PlanePath::CornerReplicate->new; my ($x, $y) = $path->n_to_xy (123);
This path is a self-similar replicating corner fill with 2x2 blocks. It's sometimes called a "U order".
7 | 63--62 59--58 47--46 43--42 | | | | | 6 | 60--61 56--57 44--45 40--41 | | | 5 | 51--50 55--54 35--34 39--38 | | | | | 4 | 48--49 52--53 32--33 36--37 | | 3 | 15--14 11--10 31--30 27--26 | | | | | 2 | 12--13 8-- 9 28--29 24--25 | | | 1 | 3-- 2 7-- 6 19--18 23--22 | | | | | Y=0 | 0-- 1 4-- 5 16--17 20--21 +-------------------------------- X=0 1 2 3 4 5 6 7
The pattern is the initial N=0 to N=3 section,
+-------+-------+ | | | | 3 | 2 | | | | +-------+-------+ | | | | 0 | 1 | | | | +-------+-------+
It then repeats as 2x2 blocks arranged in the same pattern, then 4x4 blocks, etc.
The X axis N=0,1,4,5,16,17,etc is all the integers which use only digits 0 and 1 in base 4. For example N=17 is 101 in base 4.
The Y axis N=0,3,12,15,48,etc is all the integers which use only digits 0 and 3 in base 4. For example N=51 is 303 in base 4.
And the X=Y diagonal values N=0,2,8,10,32,34,etc is all the integers which use only digits 0 and 2 in base 4.
The X axis is the same as the ZOrderCurve, and the Y axis here is the X=Y diagonal of the ZOrderCurve, and conversely the X=Y diagonal here is the Y axis of the ZOrderCurve. In general the N value at a given X,Y is converted to or from the ZOrderCurve by changing base 4 digit values 2 to 3 and 3 to 2.
A given replication extends to
Nlevel = 4^level - 1 - (2^level - 1) <= X <= (2^level - 1) - (2^level - 1) <= Y <= (2^level - 1)
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::CornerReplicate->new ()
Create and return a new path object.
($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.
$n
$n < 0
($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.
$n_lo
$n_hi
This path is in Sloane's Online Encyclopedia of Integer Sequences as
http://oeis.org/A000695 (etc) A059906 Y coordinate A000695 N on X axis, base 4 digits 0,1 only A001196 N on Y axis, base 4 digits 0,3 only A062880 N on diagonal, base 4 digits 0,2 only A163241 base-4 flip 2<->3, converts N to ZOrderCurve N (and back) A048647 permutation N at transpose Y,X base4 digits 1<->3 and 0,2 unchanged
Math::PlanePath, Math::PlanePath::LTiling, Math::PlanePath::SquareReplicate, Math::PlanePath::GosperReplicate, Math::PlanePath::ZOrderCurve, Math::PlanePath::GrayCode
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012 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/>.
To install Math::PlanePath, copy and paste the appropriate command in to your terminal.
cpanm
cpanm Math::PlanePath
CPAN shell
perl -MCPAN -e shell install Math::PlanePath
For more information on module installation, please visit the detailed CPAN module installation guide.