 Math::PlanePath::Corner(3) points shaped around in a corner

## SYNOPSIS

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

## DESCRIPTION

This path puts points in layers working outwards from the corner of the first quadrant.

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

The horizontal 1,4,9,16,etc along Y=0 is the perfect squares. This is since each further row/column ``gnomon'' added to a square makes a one-bigger square,

```                            10 11 12 13
5  6  7       5  6  7 14
2  3       2  3  8       2  3  8 15
1  4       1  4  9       1  4  9 16
2x2        3x3           4x4
```

N=2,6,12,20,etc on the diagonal X=Y-1 up from X=0,Y=1 is the pronic numbers k*(k+1) which are half way between the squares.

Each gnomon is 2 longer than the previous. This is similar to the "PyramidRows", "PyramidSides" and "SacksSpiral" paths. The "Corner" and the "PyramidSides" are the same but "PyramidSides" is stretched to two quadrants instead of one for the "Corner" here.

## Wider

An optional "wider => \$integer" makes the path wider horizontally, becoming a rectangle. For example

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

Each gnomon has the horizontal part "wider" many steps longer. Each gnomon is still 2 longer than the previous since this widening is a constant amount in each.

## 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
5  |  25 ...
4  |  16 17 18 19 20
3  |   9 10 11 12 21
2  |   4  5  6 13 22
1  |   1  2  7 14 23
Y=0  |   0  3  8 15 24
-----------------
X=0   1   2   3
```

In Nstart=0 the squares are on the Y axis and the pronic numbers are on the X=Y leading diagonal.

## FUNCTIONS

See ``FUNCTIONS'' in Math::PlanePath for behaviour common to all path classes.
"\$path = Math::PlanePath::Corner->new ()"
"\$path = Math::PlanePath::Corner->new (wider => \$w, n_start => \$n)"
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.

For "\$n < n_start()-0.5" the return is an empty list. There's an extra 0.5 before Nstart, but nothing further before there.

"\$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 as a square of side 1, so the quadrant x>=-0.5 and y>=-0.5 is entirely covered.

"(\$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.

## N to X,Y

Counting d=0 for the first L-shaped gnomon at Y=0, then the start of the gnomon is

```    StartN(d) = d^2 + 1 = 1,2,5,10,17,etc
```

The current "n_to_xy()" code extends to the left by an extra 0.5 for fractional N, so for example N=9.5 is at X=-0.5,Y=3. With this the starting N for each gnomon d is

```    StartNfrac(d) = d^2 + 0.5
```

Inverting gives the gnomon d number for an N,

```    d = floor(sqrt(N - 0.5))
```

Subtracting the gnomon start gives an offset into that gnomon

```    OffStart = N - StartNfrac(d)
```

The corner point 1,3,7,13,etc where the gnomon turns down is at d+0.5 into that remainder, and it's convenient to subtract that so negative for the horizontal and positive for the vertical,

```    Off = OffStart - (d+0.5)
= N - (d*(d+1) + 1)
```

Then the X,Y coordinates are

```    if (Off < 0)  then  X=d+Off, Y=d
if (Off >= 0) then  X=d,     Y=d-Off
```

## X,Y to N

For a given X,Y the bigger of X or Y determines the d gnomon.

If Y>=X then X,Y is on the horizontal part. At X=0 have N=StartN(d) per the Start(N) formula above, and any further X is an offset from there.

```    if Y >= X then
d=Y
N = StartN(d) + X
= Y^2 + 1 + X
```

Otherwise if Y<X then X,Y is on the vertical part. At Y=0 N is the last point on the gnomon, and one back from the start of the following gnomon,

```    if Y <= X then
d=X
LastN(d) = StartN(d+1) - 1
= (d+1)^2
N = LastN(d) - Y
= (X+1)^2 - Y
```

## Rectangle N Range

For "rect_to_n_range()", in each row increasing X is increasing N so the smallest N is in the leftmost column and the biggest N in the rightmost column.

```    |
|  ------>  N increasing
|
-----------------------
```

Going up a column, N values are increasing away from the X=Y diagonal up or down, and all N values above X=Y are bigger than the ones below.

```    |    ^  N increasing up from X=Y diagonal
|    |
|    |/
|    /
|   /|
|  / |  N increasing down from X=Y diagonal
| /  v
|/
-----------------------
```

This means the biggest N is the top right corner if that corner is Y>=X, otherwise the bottom right corner.

```                                           max N at top right
|      /                          | --+     if corner Y>=X
|     / --+                       |   | /
|    /    |                       |   |/
|   /     |                       |   |
|  /  ----v                       |  /|
| /     max N at bottom right     | --+
|/        if corner Y<=X          |/
----------                        -------
```

For the smallest N, if the bottom left corner has Y>X then it's in the ``increasing'' part and that bottom left corner is the smallest N. Otherwise Y<=X means some of the ``decreasing'' part is covered and the smallest N is at Y=min(X,Ymax), ie. either the Y=X diagonal if it's in the rectangle or the top right corner otherwise.

```    |      /
| |   /
| |  /  min N at bottom left
| +----  if corner Y>X
|  /
| /
|/
----------
|      /                           |      /
|   | /                            |     /
|   |/  min N at X=Y               |    /
|   *    if diagonal crossed       |   / +-- min N at top left
|  /|                              |  /  |    if corner Y<X
| / +-----                         | /   |
|/                                 |/
----------                         ----------
min N at Xmin,Ymin            if Ymin >= Xmin
Xmin,min(Xmin,Ymax)  if Ymin <= Xmin
```

## OEIS

This path is in Sloane's Online Encyclopedia of Integer Sequences as,

<http://oeis.org/A196199> (etc)

```    wider=0, n_start=1 (the defaults)
A213088    X+Y sum
A196199    X-Y diff, being runs -n to +n
A053615    abs(X-Y), runs n to 0 to n, distance to next pronic
A000290    N on X axis, perfect squares starting from 1
A002522    N on Y axis, Y^2+1
A002061    N on X=Y diagonal, extra initial 1
A004201    N on and below X=Y diagonal, so X>=Y
A020703    permutation N at transpose Y,X
A060734    permutation N by diagonals up from X axis
A064790     inverse
A060736    permutation N by diagonals down from Y axis
A064788     inverse
A027709    boundary length of N unit squares
A078633    grid sticks of N points
n_start=0
A000196    max(X,Y), being floor(sqrt(N))
A005563    N on X axis, n*(n+2)
A000290    N on Y axis, perfect squares
A002378    N on X=Y diagonal, pronic numbers
n_start=2
A059100    N on Y axis, Y^2+2
A014206    N on X=Y diagonal, pronic+2
wider=1
A053188    abs(X-Y), dist to nearest square, extra initial 0
wider=1, n_start=0
A002378    N on Y axis, pronic numbers
A005563    N on X=Y diagonal, n*(n+2)
wider=1, n_start=2
A014206    N on Y axis, pronic+2
wider=2, n_start=0
A005563    N on Y axis, (Y+1)^2-1
A028552    N on X=Y diagonal, k*(k+3)
wider=3, n_start=0
A028552    N on Y axis, k*(k+3)
```

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