Math::PlanePath::Corner -- points shaped in a corner
use Math::PlanePath::Corner; my $path = Math::PlanePath::Corner->new; my ($x, $y) = $path->n_to_xy (123);
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 are the perfect squares, which is simply because 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
The diagonal 2,6,12,20,etc upwards from X=0,Y=1 are 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.
$path = Math::PlanePath::Corner->new ()
Create and return a new path object.
($x,$y) = $path->n_to_xy ($n)
Return the x,y coordinates of point number
$non the path.
$n < 0.5the 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
$yare 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.
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
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
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
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.
Math-PlanePath is Copyright 2010, 2011 Kevin Ryde
Math-PlanePath 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.
Math-PlanePath 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.
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