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# NAME

Math::PlanePath::Corner -- 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 stripe 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``````

The diagonal 2,6,12,20,etc upwards from X=0,Y=1 is the pronic numbers k*(k+1), half way between those squares.

Each row/column stripe is 2 longer than the previous, similar to the PyramidRows, PyramidSides and SacksSpiral paths. The Corner and the PyramidSides are the same, just the PyramidSides stretched out to two quadrants instead of one for this Corner.

## Wider

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

``    \$path = Math::PlanePath::Corner->new (wider => 3);``

gives

``````     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``````

The initial N=1 is `wider` many further places to the right before going up to the Y axis, then the path makes corners around that shape.

Each loop is still 2 longer than the previous, as the widening is a constant amount in each loop.

# FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.

`\$path = Math::PlanePath::Corner->new ()`
`\$path = Math::PlanePath::Corner->new (wider => \$w)`

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 < 0.5` the return is an empty list, it being considered there are no points before 1 in the corner.

`\$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.

# FORMULAS

## N to X,Y

Counting d=0 for the first row at y=0, then the start of that row N=1,2,5,10,17,etc is

``    StartN(d) = d^2 + 1``

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 d row is

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

Inverting gives the row for an N,

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

And subtracting that start gives an offset into the row

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

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

``````    Rem = RemStart - (d+0.5)
= N - (d*(d+1) + 1)``````

And the x,y coordinates thus

``````    if (Rem < 0)  then x=d+Rem, y=d
if (Rem >= 0) then x=d, y=d-Rem``````

## X,Y to N

For a given x,y the bigger of x or y determines the d row. If y>=x then x,y is on the horizontal part with d=y and in that case StartN(d) above is the N for x=0, and the given x can be added to that,

``````    N = StartN(d) + x
= y^2 + 1 + x``````

Or otherwise if y<x then x,y is on the vertical and d=x. In that case the y=0 is the last point on the row and is one back from the start of the following row,

``````    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 in the rightmost.

Going up a column, N values decrease until reaching X=Y, and then increase, with those values above X=Y all bigger than the ones below. This means the biggest N is the top right corner if it has Y>=X, otherwise the bottom right corner.

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.

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