| Math::PlanePath::CornerAlternating(3) | User Contributed Perl Documentation | Math::PlanePath::CornerAlternating(3) |
NAME¶
Math::PlanePath::CornerAlternating -- points shaped around a corner alternately
SYNOPSIS¶
use Math::PlanePath::CornerAlternating; my $path = Math::PlanePath::CornerAlternating->new; my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION¶
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.
Wider¶
An optional "wider => $integer" makes the path wider horizontally, becoming a rectangle. For example
4 | 29--30--31--32--33--34--35--36 ...
| | | |
3 | 28--27--26--25--24--23--22 37 44 wider => 3
| | | |
2 | 11--12--13--14--15--16 21 38 43
| | | | | |
1 | 10---9---8---7---6 17 20 39 42
| | | | | |
Y=0 | 1---2---3---4---5 18--19 40--41
+--------------------------------------
X=0 1 2 3 4 5 6 7 8
Each gnomon has the horizontal part "wider" many steps longer. For wider=3 shown, the additional points are 2,3,4 in the first row, then 5..10 are the next gnomon. Each gnomon is still 2 longer than the previous since this widening is a constant amount in each.
N Start¶
The default is to number points starting N=1 as shown above. An optional "n_start" can give a different start with the same shape etc. For example to start at 0,
4 | 16 17 18 19 20
3 | 15 14 13 12 21 n_start => 0
2 | 4 5 6 11 22
1 | 3 2 7 10 23
Y=0 | 0 1 8 9 24
---------------------
X=0 1 2 3 4
With Nstart=0, the pronic numbers are on the X=Y leading diagonal.
FUNCTIONS¶
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
- "$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 $n on
the path.
For "$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 = $path->xy_to_n ($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¶
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.
Rectangle N Range¶
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,
| 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.
OEIS¶
This path is in Sloane's Online Encyclopedia of Integer Sequences as,
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
SEE ALSO¶
Math::PlanePath, Math::PlanePath::Corner, Math::PlanePath::DiagonalsAlternating
HOME PAGE¶
LICENSE¶
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/>.
| 2021-01-20 | perl v5.40.0 |