 Math::PlanePath::HeptSpiralSkewed(3) integer points around a skewed seven sided spiral

## SYNOPSIS

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

## DESCRIPTION

This path makes a seven-sided spiral by cutting one corner of a square

```    31-30-29-28                       3
|         \
32 14-13-12 27                    2
|  |      \  \
33 15  4--3 11 26                 1
|  |  |   \  \  \
34 16  5  1--2 10 25         <- Y=0
|  |  |        |  |
35 17  6--7--8--9 24             -1
|  |              |
36 18-19-20-21-22-23             -2
|
37-38-39-40-41-...               -3
^
-3 -2 -1 X=0 1  2  3
```

The path is as if around a heptagon, with the left and bottom here as two sides of the heptagon straightened out, and the flat top here skewed across to fit a square grid.

## N Start

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

```    30 29 28 27              n_start => 0
31 13 12 11 26
32 14  3  2 10 25
33 15  4  0  1  9 24
34 16  5  6  7  8 23
35 17 18 19 20 21 22
36 37 38 39 40 ...
```

## FUNCTIONS

See ``FUNCTIONS'' in Math::PlanePath for behaviour common to all path classes.
"\$path = Math::PlanePath::HeptSpiralSkewed->new ()"
"\$path = Math::PlanePath::HeptSpiralSkewed->new (n_start => \$n)"
Create and return a new path object.
"\$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 N in the path as centred in a square of side 1, so the entire plane is covered.

## N to X,Y

It's convenient to work in terms of Nstart=0 and to take each loop as beginning on the South-West diagonal,

```              top length = d
30-29-28-27
|         \
31          26    diagonal length = d
left        |            \
length     32             25
= 2*d      |               \
33        0       24
|                 |    right
34     .          23    length = d-1
|                 |
35 17-18-19-20-21-22
|
.    bottom length = 2*d-1
```

The SW diagonal is N=0,5,17,36,etc which is

```    N = (7d-11)*d/2 + 2           # starting d=1 first loop
```

This can be inverted to get d from N

```    d = floor( (sqrt(56*N+9)+11)/14 )
```

The side lengths are as shown above. The first loop is d=1 and for it the ``right'' vertical length is zero, so no such side on that first loop 0 <= N < 5.

## OEIS

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

<http://oeis.org/A192136> (etc)

```    n_start=1
A140065    N on Y axis
n_start=0
A001106    N on X axis, 9-gonal numbers
A218471    N on Y axis
A022265    N on X negative axis
A179986    N on Y negative axis, second 9-gonals
A195023    N on X=Y diagonal
A022264    N on North-West diagonal
A186029    N on South-West diagonal
A024966    N on South-East diagonal
```

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