/ 
Math-PlanePath-129
River stage one • 3 direct dependents • 4 total dependents
/ Math::PlanePath::CincoCurve

NAME

Math::PlanePath::CincoCurve -- 5x5 self-similar curve

SYNOPSIS

use Math::PlanePath::CincoCurve;
my $path = Math::PlanePath::CincoCurve->new;
my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

This is the 5x5 self-similar Cinco curve

It makes a 5x5 self-similar traversal of the first quadrant X>0,Y>0.

                                                |
  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.

Dennis conceives 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 top-level target size rather than the unlimited first quadrant here.

FUNCTIONS

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.

Level Methods

($n_lo, $n_hi) = $path->level_to_n_range($level)

Return (0, 25**$level - 1).

SEE ALSO

Math::PlanePath, Math::PlanePath::PeanoCurve, Math::PlanePath::DekkingCentres

HOME PAGE

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

LICENSE

Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 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.

You should have received a copy of the GNU General Public License along with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.