++ed by:
Kevin Ryde
and 1 contributors

# NAME

Math::PlanePath::WunderlichMeander -- 3x3 self-similar "R" shape

# SYNOPSIS

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

# DESCRIPTION

This is an integer version of the 3x3 self-similar meander from

``````      8     20--21--22  29--30--31  38--39--40
|       |   |       |   |       |
7     19  24--23  28  33--32  37  42--41
|   |       |   |       |   |
6     18  25--26--27  34--35--36  43--44
|                               |
5     17  14--13  56--55--54--53--52  45
|   |   |   |               |   |
4     16--15  12  57  60--61  50--51  46
|   |   |   |   |       |
3      9--10--11  58--59  62  49--48--47
|                   |
2      8   5-- 4  65--64--63  74--75--76
|   |   |   |           |       |
1      7-- 6   3  66  69--70  73  78--77
|   |   |   |   |   |
Y=0->    0-- 1-- 2  67--68  71--72  79--80-...

X=0  1   2   3   4   5   6   7   8``````

The base pattern is the N=0 to N=8 section. It works as a traversal of a 3x3 square starting in one corner and going along one side. The base figure goes upwards and it's then used rotated by 180 degrees and/or transposed to go in other directions,

``````    +----------------+----------------+---------------+
| ^              |              * | ^             |
| |              |  rotate 180  | | |   base      |
| |     8        |       5      | | |     4       |
| |   base       |              | | |             |
| *              |              v | *             |
+----------------+----------------+---------------+
| <------------* | <------------* | ^             |
|                |                | |             |
|       7        |       6        | |     3       |
|   rotate 180   |   rotate 180   | |   base      |
|  + transpose   |  + transpose   | *             |
+----------------+----------------+---------------+
|                |                | ^             |
|                |                | |             |
|       0        |       1        | |     2       |
|   transpose    |   transpose    | |   base      |
| *----------->  | *------------> | *             |
+----------------+----------------+---------------+``````

The base 0 to 8 goes upwards, so the across sub-parts are an X,Y transpose. The transpose in the 0 part means the higher levels go alternately up or across. So N=0 to N=8 goes up, then the next level N=0,9,18,.,72 goes right, then N=81,162,..,648 up again, etc.

Wunderlich's conception is successive lower levels of detail as a space-filling curve. The transposing in that case applies to ever smaller parts. But for the integer version here, the start direction is held fixed and the successively higher levels alternate. The first move N=0 to N=1 is rightwards per the "Schema" shown in Wunderlich's paper (and like various other `PlanePath` curves).

# FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

`\$path = Math::PlanePath::WunderlichMeander->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_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.

## Level Methods

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

Return `(0, 9**\$level - 1)`.

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