Math::PlanePath::QuintetCurve(3) self-similar plus shaped curve

## SYNOPSIS

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

## DESCRIPTION

This path is traces out a spiralling self-similar ``+'' shape,

```            125--...                 93--92                      11
|                       |   |
123-124                      94  91--90--89--88          10
|                           |               |
122-121-120 103-102          95  82--83  86--87           9
|   |   |           |   |   |   |
115-116 119 104 101-100--99  96  81  84--85               8
|   |   |   |           |   |   |
113-114 117-118 105  32--33  98--97  80--79--78               7
|               |   |   |                   |
112-111-110-109 106  31  34--35--36--37  76--77               6
|   |   |               |   |
108-107  30  43--42  39--38  75                   5
|   |   |   |       |
25--26  29  44  41--40  73--74                   4
|   |   |   |           |
23--24  27--28  45--46--47  72--71--70--69--68       3
|                       |                   |
22--21--20--19--18  49--48  55--56--57  66--67       2
|   |       |       |   |
5---6---7  16--17  50--51  54  59--58  65           1
|       |   |           |   |   |       |
0---1   4   9---8  15          52--53  60--61  64       <- Y=0
|   |   |       |                       |   |
2---3  10--11  14                      62--63          -1
|   |
12--13                                      -2
^
X=0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 ...
```

The base figure is the initial N=0 to N=4.

```              5
|
|
0---1   4      base figure
|   |
|   |
2---3
```

It corresponds to a traversal of the following ``+'' shape,

```         .... 5
.    |
.   <|
|
0----1 .. 4 ....
v  |    |    .
.    |>   |>   .
.    |    |    .
.... 2----3 ....
. v  .
.    .
.    .
. .. .
```

The ``v'' and ``>'' notches are the side the figure is directed at the higher replications. The 0, 2 and 3 parts are the right hand side of the line and are a plain repetition of the base figure. The 1 and 4 parts are to the left and are a reversal. The first such reversal is seen above as N=5 to N=10.
.....
.   .

```    5---6---7 ...
.   .   |   .
.       |   .   reversed figure
... 9---8 ...
|   .
|   .
10 ...
```

In the base figure it can be seen the N=5 endpoint is rotated up around from the N=0 to N=1 direction. This makes successive higher levels slowly spiral around.

```    N = 5^level
angle = level * atan(1/2)
= level * 26.56 degrees
```

In the sample shown above N=125 is level=3 and has spiralled around to angle 3*26.56=79.7 degrees. The next level goes into the second quadrant with X negative. A full circle around the plane is around level 14.

## Arms

The optional "arms => \$a" parameter can give 1 to 4 copies of the curve, each advancing successively. For example "arms=>4" is as follows. N=4*k points are the plain curve, and N=4*k+1, N=4*k+2 and N=4*k+3 are rotated copies of it.

```                    69--65                      ...
|   |                       |
..-117-113-109  73  61--57--53--49         120
|   |               |           |
101-105  77  25--29  41--45 100-104 116
|       |   |   |   |       |   |   |
97--93  81  21  33--37  92--96 108-112
|   |   |           |
50--46  89--85  17--13-- 9  88--84--80--76--72
|   |                   |                   |
54  42--38  10-- 6   1-- 5  20--24--28  64--68
|       |   |   |           |       |   |
58  30--34  14   2   0-- 4  16  36--32  60
|   |       |           |   |   |       |
66--62  26--22--18   7-- 3   8--12  40--44  56
|                   |                   |   |
70--74--78--82--86  11--15--19  87--91  48--52
|           |   |   |
110-106  94--90  39--35  23  83  95--99
|   |   |       |   |   |   |       |
114 102--98  47--43  31--27  79 107-103
|           |               |   |
118          51--55--59--63  75 111-115-119-..
|                       |   |
...                     67--71
```

The curve is essentially an ever expanding ``+'' shape with one corner at the origin. Four such shapes can be packed as follows,

```                +---+
|   |
+---+---    +---+
|   |     A     |
+---+   +---+   +---+
|     B     |   |   |
+---+   +---O---+   +---+
|   |   |     D     |
+---+   +---+   +---+
|     C     |   |
+---+   +---+---+
|   |
+---+
```

At higher replication levels the sides are wiggly and spiralling and the centres of each rotated around, but they sides are symmetric and mesh together perfectly to fill the plane.

## FUNCTIONS

See ``FUNCTIONS'' in Math::PlanePath for behaviour common to all path classes.
"\$path = Math::PlanePath::QuintetCurve->new ()"
"\$path = Math::PlanePath::QuintetCurve->new (arms => \$a)"
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.

Fractional positions give an X,Y position along a straight line between the integer positions.

"\$n = \$path->n_start()"
Return 0, the first N in the path.
"(\$n_lo, \$n_hi) = \$path->rect_to_n_range (\$x1,\$y1, \$x2,\$y2)"
In the current code the returned range is exact, meaning \$n_lo and \$n_hi are the smallest and biggest in the rectangle, but don't rely on that yet since finding the exact range is a touch on the slow side. (The advantage of which though is that it helps avoid very big ranges from a simple over-estimate.)

## Level Methods

"(\$n_lo, \$n_hi) = \$path->level_to_n_range(\$level)"
Return "(0, 5**\$level)", or for multiple arms return "(0, \$arms * 5**\$level)".

There are 5^level + 1 points in a level, numbered starting from 0. On the second and subsequent arms the origin is omitted (so as not to repeat that point) and so just 5^level for them, giving 5^level+1 + (arms-1)*5^level = arms*5^level + 1 many points starting from 0.

## X,Y to N

The current approach uses the "QuintetCentres" "xy_to_n()". Because the tiling in "QuintetCurve" and "QuintetCentres" is the same, the X,Y coordinates for a given N are no more than 1 away in the grid.

The way the two lowest shapes are arranged in fact means that for a "QuintetCurve" N at X,Y then the same N on the "QuintetCentres" is at one of three locations

```    X, Y          same
X, Y+1        up
X-1, Y+1      up and left
X-1, Y        left
```

This is so even when the ``arms'' multiple paths are in use (the same arms in both coordinates).

Is there an easy way to know which of the four offsets is right? The current approach is to give each to "QuintetCentres" to make an N, put that N back through "n_to_xy()" to see if it's the target \$n.

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