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  1718192021   3  10111213 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 onebigger 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=Y1 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 => $integer" makes the path wider horizontally, becoming a rectangle. For example
wider => 3 4  293031...  3  19202122232425   2  111213141516 26    1  56789 17 27     Y=0  1234 10 18 28   ^ X=0 1 2 3 4 5 6
Each gnomon has the horizontal part "wider" many steps longer. Each gnomon is still 2 longer than the previous since this widening is a constant amount in each.
N Start
The default is to number points starting N=1 as shown above. An optional "n_start" can give a different start with the same shape etc. For example to start at 0,
n_start => 0 5  25 ... 4  16 17 18 19 20 3  9 10 11 12 21 2  4 5 6 13 22 1  1 2 7 14 23 Y=0  0 3 8 15 24  X=0 1 2 3
In Nstart=0 the squares are on the Y axis and the pronic numbers are on the X=Y leading diagonal.
FUNCTIONS
See ``FUNCTIONS'' in Math::PlanePath for behaviour common to all path classes. "$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 Lshaped 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=dOff
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 XY diff, being runs n to +n A053615 abs(XY), 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(XY), 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)^21 A028552 N on X=Y diagonal, k*(k+3) wider=3, n_start=0 A028552 N on Y axis, k*(k+3)
LICENSE
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016 Kevin RydeThis file is part of MathPlanePath.
MathPlanePath is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.
MathPlanePath is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
You should have received a copy of the GNU General Public License along with MathPlanePath. If not, see <http://www.gnu.org/licenses/>.