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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 "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.

FORMULAS

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,

``````    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