 Math::PlanePath::TriangleSpiralSkewed(3) integer points drawn around a skewed equilateral triangle

## SYNOPSIS

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

## DESCRIPTION

This path makes an spiral shaped as an equilateral triangle (each side the same length), but skewed to the left to fit on a square grid,

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

The properties are the same as the spread-out "TriangleSpiral". The triangle numbers fall on straight lines as the do in the "TriangleSpiral" but the skew means the top corner goes up at an angle to the vertical and the left and right downwards are different angles plotted (but are symmetric by N count).

## Skew Right

Option "skew => 'right'" directs the skew towards the right, giving

```      4                  16      skew="right"
/ |
3               17 15
/    |
2            18  4 14
/  / |  |
1        ...  5  3 13
/    |  |
Y=0 ->       6  1--2 12
/          |
-1       7--8--9-10-11
^
-2 -1 X=0 1  2
```

This is a shear ``X -> X+Y'' of the default skew=``left'' shown above. The coordinates are related by

```    Xright = Xleft + Yleft         Xleft = Xright - Yright
Yright = Yleft                 Yleft = Yright
```

## Skew Up

```      2       16-15-14-13-12-11      skew="up"
|            /
1       17  4--3--2 10
|  |   /  /
Y=0 ->    18  5  1  9
|  |   /
-1      ...  6  8
|/
-2           7
^
-2 -1 X=0 1  2
```

This is a shear ``Y -> X+Y'' of the default skew=``left'' shown above. The coordinates are related by

```    Xup = Xleft                 Xleft = Xup
Yup = Yleft + Xleft         Yleft = Yup - Xup
```

## Skew Down

```      2          ..-18-17-16       skew="down"
|
1        7--6--5--4 15
\       |  |
Y=0 ->        8  1  3 14
\  \ |  |
-1              9  2 13
\    |
-2                10 12
\ |
11
^
-2 -1 X=0 1  2
```

This is a rotate by -90 degrees of the skew=``up'' above. The coordinates are related

```    Xdown = Yup          Xup = - Ydown
Ydown = - Xup        Yup = Xdown
```

Or related to the default skew=``left'' by

```    Xdown = Yleft + Xleft        Xleft = - Ydown
Ydown = - Xleft              Yleft = Xdown + Ydown
```

## 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,

```    15        n_start => 0
|\
16 14
|   \
17  3 13 ...
|  |\  \  \
18  4  2 12 31
|  |   \  \  \
19  5  0--1 11 30
|  |         \  \
20  6--7--8--9-10 29
|                  \
21-22-23-24-25-26-27-28
```

With this adjustment for example the X axis N=0,1,11,30,etc is (9X-7)*X/2, the hendecagonal numbers (11-gonals). And South-East N=0,8,25,etc is the hendecagonals of the second kind, (9Y-7)*Y/2 with Y negative.

## FUNCTIONS

See ``FUNCTIONS'' in Math::PlanePath for behaviour common to all path classes.
"\$path = Math::PlanePath::TriangleSpiralSkewed->new ()"
"\$path = Math::PlanePath::TriangleSpiralSkewed->new (skew => \$str, n_start => \$n)"
Create and return a new skewed triangle spiral object. The "skew" parameter can be

```    "left"    (the default)
"right"
"up"
"down"
```
"\$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.

## Rectangle to N Range

Within each row there's a minimum N and the N values then increase monotonically away from that minimum point. Likewise in each column. This means in a rectangle the maximum N is at one of the four corners of the rectangle.

```              |
x1,y2 M---|----M x2,y2        maximum N at one of
|   |    |              the four corners
-------O---------          of the rectangle
|   |    |
|   |    |
x1,y1 M---|----M x1,y1
|
```

## OEIS

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

<http://oeis.org/A117625> (etc)

```    n_start=1, skew="left" (the defaults)
A204439     abs(dX)
A204437     abs(dY)
A010054     turn 1=left,0=straight, extra initial 1
A117625     N on X axis
A064226     N on Y axis, but without initial value=1
A006137     N on X negative
A064225     N on Y negative
A081589     N on X=Y leading diagonal
A038764     N on X=Y negative South-West diagonal
A081267     N on X=-Y negative South-East diagonal
A060544     N on ESE slope dX=+2,dY=-1
A081272     N on SSE slope dX=+1,dY=-2
A217010     permutation N values of points in SquareSpiral order
A217291      inverse
A214230     sum of 8 surrounding N
A214231     sum of 4 surrounding N
n_start=0
A051682     N on X axis (11-gonal numbers)
A081268     N on X=1 vertical (next to Y axis)
A062708     N on Y axis
A062725     N on Y negative axis
A081275     N on X=Y+1 North-East diagonal
A062728     N on South-East diagonal (11-gonal second kind)
A081266     N on X=Y negative South-West diagonal
A081270     N on X=1-Y North-West diagonal, starting N=3
A081271     N on dX=-1,dY=2 NNW slope up from N=1 at X=1,Y=0
n_start=-1
A023531     turn 1=left,0=straight, being 1 at N=k*(k+3)/2
A023532     turn 1=straight,0=left
n_start=1, skew="right"
A204435     abs(dX)
A204437     abs(dY)
A217011     permutation N values of points in SquareSpiral order
but with 90-degree rotation
A217292     inverse
A214251     sum of 8 surrounding N
n_start=1, skew="up"
A204439     abs(dX)
A204435     abs(dY)
A217012     permutation N values of points in SquareSpiral order
but with 90-degree rotation
A217293     inverse
A214252     sum of 8 surrounding N
n_start=1, skew="down"
A204435     abs(dX)
A204439     abs(dY)
```

The square spiral order in A217011,A217012 and their inverses has first step at 90-degrees to the first step of the triangle spiral, hence the rotation by 90 degrees when relating to the "SquareSpiral" path. A217010 on the other hand has no such rotation since it reckons the square and triangle spirals starting in the same direction.

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