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 4The 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 4x4The 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 6The 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.
N Start
The default is to number points starting N=1 as shown above. An optional n_start can give a different start, in the same sequence. 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 3FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for 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
$non the path.For
$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
$x,$y.$xand$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. ($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)-
The returned range is exact, meaning
$n_loand$n_hiare the smallest and biggest in the rectangle.
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 + 1The 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 + 0.5Inverting 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-RemX,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 + XOr 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 - YRectangle N Range
For rect_to_n_range(), in each row X increasing is N increasing 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 the 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.
OEIS
This path is in Sloane's Online Encyclopedia of Integer Sequences as,
http://oeis.org/A196199 (etc)
wider=0 (the default)
A196199 X-Y, being runs -n to +n
A053615 abs(X-Y), distance to next pronic
A002522 N on Y axis (N=Y^2+1)
A020703 permutation N at transpose Y,X
wider=0,n_start=0
A005563 N on X_axis, (X+1)^2-1
A000290 N on Y axis, perfect squares
A002378 N on X=Y diagonal, pronic
wider=1
A053188 abs(X-Y), dist to nearest square, extra initial 0
wider=1,n_start=0
A002378 N on Y_axis, pronic
A005563 N on X=Y diagonal, (k+1)^2-1
wider=2,n_start=0
A005563 N on Y axis, (Y+1)^2-1SEE ALSO
Math::PlanePath, Math::PlanePath::PyramidRows, Math::PlanePath::PyramidSides, Math::PlanePath::SacksSpiral, Math::PlanePath::Diagonals
HOME PAGE
http://user42.tuxfamily.org/math-planepath/index.html
LICENSE
Copyright 2010, 2011, 2012 Kevin Ryde
This file is part of Math-PlanePath.
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.
You should have received a copy of the GNU General Public License along with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.