---
name: "Math::PlanePath::CincoCurve"
section: "3pm"
platform: "linux"
package: "libmath-planepath-perl"
version: "129-1"
release: "Debian 13 (Trixie)"
platforms: ["linux"]
url: "https://new.carta.tech/Math::PlanePath::CincoCurve"
---

# Math::PlanePath::CincoCurve(3pm)

## 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

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\>

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/\>.