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
$path = Math::PlanePath::Corner->new (wider => 3);
gives
4 | 29--30--31--...
|
3 | 19--20--21--22--23--24--25
| |
2 | 11--12--13--14--15--16 26
| | |
1 | 5---6---7---8---9 17 27
| | | |
Y=0 | 1---2---3---4 10 18 28
|
-----------------------------
^
X=0 1 2 3 4 5 6
Each gnomon has the horizontal part "wider" many steps longer. Each
gnomon is still 2 longer than the previous since this widening is a constant
amount in each.
N Start¶
The default is to number points starting N=1 as shown above. An optional
"n_start" can give a different start with the same shape etc. For
example to start at 0,
n_start => 0
5 | 25 ...
4 | 16 17 18 19 20
3 | 9 10 11 12 21
2 | 4 5 6 13 22
1 | 1 2 7 14 23
Y=0 | 0 3 8 15 24
-----------------
X=0 1 2 3
In Nstart=0 the squares are on the Y axis and the pronic numbers are on the X=Y
leading diagonal.
FUNCTIONS¶
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
classes.
- "$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.
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)
SEE ALSO¶
Math::PlanePath, Math::PlanePath::PyramidSides, Math::PlanePath::PyramidRows,
Math::PlanePath::SacksSpiral, Math::PlanePath::Diagonals
HOME PAGE¶
<
http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE¶
Copyright 2010, 2011, 2012, 2013, 2014 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/>.