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.

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/>.