Math::PlanePath::CornerAlternating -- points shaped around a corner alternately
use Math::PlanePath::CornerAlternating; my $path = Math::PlanePath::CornerAlternating->new; my ($x, $y) = $path->n_to_xy (123);
This path is points in layers around a square outwards from a corner in the first quadrant, alternately upward or downward. Each row/column "gnomon" added to a square makes a one-bigger square.
4 | 17--18--19--20--21 ... | | | | 3 | 16--15--14--13 22 29 | | | | 2 | 5---6---7 12 23 28 | | | | | | 1 | 4---3 8 11 24 27 | | | | | | Y=0 | 1---2 9--10 25--26 +------------------------- X=0 1 2 3 4 5
This is like the Corner path, but here gnomons go back and forward and in particular so points are always a unit step apart.
An optional wider => $integer makes the path wider horizontally, becoming a rectangle. For example
wider => $integer
4 | 29--30--31--32--33--34--35--36 ... | | | | 3 | 28--27--26--25--24--23--22 37 44 wider => 3 | | | | 2 | 11--12--13--14--15--16 21 38 43 | | | | | | 1 | 10---9---8---7---6 17 20 39 42 | | | | | | Y=0 | 1---2---3---4---5 18--19 40--41 +-------------------------------------- X=0 1 2 3 4 5 6 7 8
Each gnomon has the horizontal part wider many steps longer. For wider=3 shown, the additional points are 2,3,4 in the first row, then 5..10 are the next gnomon. Each gnomon is still 2 longer than the previous since this widening is a constant amount in each.
wider
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
4 | 16 17 18 19 20 3 | 15 14 13 12 21 n_start => 0 2 | 4 5 6 11 22 1 | 3 2 7 10 23 Y=0 | 0 1 8 9 24 --------------------- X=0 1 2 3 4
With Nstart=0, the pronic numbers are on the X=Y leading diagonal.
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::CornerAlternating->new ()
$path = Math::PlanePath::CornerAlternating->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.
$n
For $n < n_start() the return is an empty list. Fractional $n gives an X,Y position along a straight line between the integer positions.
$n < n_start()
$n = $path->xy_to_n ($x,$y)
Return the point number for coordinates $x,$y.
$x,$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.
$x
$y
($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_lo
$n_hi
Most calculations are similar to the Corner path (without the 0.5 fractional part), and a reversal applied when the d gnomon number is odd. When wider>0, that reversal must allow for the horizontals and verticals different lengths.
For rect_to_n_range(), the largest gnomon is either the top or right of the rectangle, depending where the top right corner x2,y2 falls relative to the leading diagonal,
rect_to_n_range()
| A---B / x2<y2 | / x2>y2 | | |/ top | +------B right | | | row | | / | side | | /| biggest | | / | biggest | +---+ gnomon | +------C gnomon | / | / +--------- +-----------
Then the maximum is at A or B, or B or C according as which way that gnomon goes, so odd or even.
If it happens that B is on the diagonal, so x2=y2, then it's either A or C according as the gnomon odd or even
| / | A----+ x2=y2 | | /| | | / | | +----C | / +-----------
For wider > 0, the diagonal shifts across so that x2-wider <=> y2 is the relevant test.
This path is in Sloane's Online Encyclopedia of Integer Sequences as,
http://oeis.org/A319289 (etc)
wider=0, n_start=1 (the defaults) A220603 X+1 coordinate A220604 Y+1 coordinate A213088 X+Y sum A081346 N on X axis A081345 N on Y axis A002061 N on X=Y diagonal, extra initial 1 A081344 permutation N by diagonals A194280 inverse A020703 permutation N at transpose Y,X A027709 boundary length of N unit squares A078633 grid sticks of N points n_start=0 A319290 X coordinate A319289 Y coordinate A319514 Y,X coordinate pairs A329116 X-Y diff A053615 abs(X-Y) diff A000196 max(X,Y), being floor(sqrt(N)) A339265 dX-dY increments (runs +1,-1) A002378 N on X=Y diagonal, pronic numbers A220516 permutation N by diagonals n_start=2 A014206 N on X=Y diagonal, pronic+2 wider=1, n_start=1 A081347 N on X axis A081348 N on Y axis A080335 N on X=Y diagonal A093650 permutation N by diagonals wider=1, n_start=0 A180714 X-Y diff wider=2, n_start=1 A081350 N on X axis A081351 N on Y axis A081352 N on X=Y diagonal A081349 permutation N by diagonals
Math::PlanePath, Math::PlanePath::Corner, Math::PlanePath::DiagonalsAlternating
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2021 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/>.
To install Math::PlanePath, copy and paste the appropriate command in to your terminal.
cpanm
cpanm Math::PlanePath
CPAN shell
perl -MCPAN -e shell install Math::PlanePath
For more information on module installation, please visit the detailed CPAN module installation guide.