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,

<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

HOME PAGE

<http://user42.tuxfamily.org/math-planepath/index.html>

LICENSE

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/>.

2021-01-23

perl v5.32.0