 Math::PlanePath::HexSpiralSkewed(3) integer points around a skewed hexagonal spiral

## SYNOPSIS

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

## DESCRIPTION

This path makes a hexagonal spiral with points skewed so as to fit a square grid and fully cover the plane.

```    13--12--11   ...              2
|         \   \
14   4---3  10  23            1
|   |     \   \   \
15   5   1---2   9  22    <- Y=0
\   \          |   |
16   6---7---8  21       -1
\              |
17--18--19--20       -2
^   ^   ^   ^   ^   ^
-2  -1  X=0  1   2   3  ...
```

The kinds of N=3*k^2 numbers which fall on straight lines in the plain "HexSpiral" also fall on straight lines when skewed. See Math::PlanePath::HexSpiral for notes on this.

## Skew

The skewed path is the same shape as the plain "HexSpiral", but fits more points on a square grid. The skew pushes the top horizontal to the left, as shown by the following parts, and the bottom horizontal is similarly skewed but to the right.

```    HexSpiralSkewed               HexSpiral
13--12--11                   13--12--11
|         \                /          \
14          10            14            10
|             \         /                \
15               9     15                   9
-2  -1  X=0  1   2     -4 -3 -2  X=0  2  3  4
```

In general the coordinates can be converted each way by

```    plain X,Y -> skewed (X-Y)/2, Y
skewed X,Y -> plain 2*X+Y, Y
```

## Corners

"HexSpiralSkewed" is similar to the "SquareSpiral" but cuts off the top-right and bottom-left corners so that each loop is 6 steps longer than the previous, whereas for the "SquareSpiral" it's 8. See ``Corners'' in Math::PlanePath::SquareSpiral for other corner cutting.

## Wider

An optional "wider" parameter makes the path wider, stretched along the top and bottom horizontals. For example

```    \$path = Math::PlanePath::HexSpiralSkewed->new (wider => 2);
```

gives

```    21--20--19--18--17                    2
|                 \
22   8---7---6---5  16                1
|   |             \   \
23   9   1---2---3---4  15        <- Y=0
\   \                  |
24   10--11--12--13--14  ...      -1
\                      |
25--26--27--28--29--30       -2
^   ^   ^   ^   ^   ^   ^   ^
-4  -3  -2  -1  X=0  1   2   3  ...
```

The centre horizontal from N=1 is extended by "wider" many further places, then the path loops around that shape. The starting point 1 is shifted to the left by wider/2 places (rounded up to an integer) to keep the spiral centred on the origin X=0,Y=0.

Each loop is still 6 longer than the previous, since the widening is basically a constant amount added into each loop. The result is the same as the plain "HexSpiral" of the same widening too. The effect looks better in the plain "HexSpiral".

## N Start

The default is to number points starting N=1 as shown above. An optional "n_start" can give a different start with the same shape etc. For example to start at 0,

```    n_start => 0
27  26  25  24                            3
28  12  11  10  23                        2
29  13   3   2   9  22                    1
30  14   4   0   1   8  21 ...       <- Y=0
31  15   5   6   7  20  39           -1
32  16  17  18  19  38           -2
33  34  35  36  37           -3

-3  -2  -1  X=0  1   2   3   4
```

In this numbering the X axis N=0,1,8,21,etc is the octagonal numbers 3*X*(X+1).

## FUNCTIONS

See ``FUNCTIONS'' in Math::PlanePath for behaviour common to all path classes.
"\$path = Math::PlanePath::HexSpiralSkewed->new ()"
"\$path = Math::PlanePath::HexSpiralSkewed->new (wider => \$w)"
Create and return a new hexagon spiral object. An optional "wider" parameter widens the spiral path, it defaults to 0 which is no widening.
"\$n = \$path->xy_to_n (\$x,\$y)"
Return the point number for coordinates "\$x,\$y". \$x and \$y are each rounded to the nearest integer, which has the effect of treating each point in the path as a square of side 1.

## OEIS

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

<http://oeis.org/A056105> (etc)

```    A056105    N on X axis, 3n^2-2n+1
A056106    N on Y axis, 3n^2-n+1
A056107    N on North-West diagonal, 3n^2+1
A056108    N on X negative axis, 3n^2+n+1
A056109    N on Y negative axis, 3n^2+2n+1
A003215    N on South-East diagonal, centred hexagonals
n_start=0
A000567    N on X axis, octagonal numbers
A049450    N on Y axis
A049451    N on X negative axis
A045944    N on Y negative axis, octagonal numbers second kind
A062783    N on X=Y diagonal north-east
A033428    N on north-west diagonal, 3*k^2
A063436    N on south-west diagonal
A028896    N on south-east diagonal
```

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