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  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)
SEE ALSO
Math::PlanePath, Math::PlanePath::PyramidSides, Math::PlanePath::PyramidRows, Math::PlanePath::SacksSpiral, Math::PlanePath::Diagonals
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017 Kevin Ryde
This 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/>.