Reference

Math::PlanePath::DiagonalsAlternating(3pm)

NAME

Math::PlanePath::DiagonalsAlternating -- points in diagonal stripes of alternating directions

SYNOPSIS

 use Math::PlanePath::DiagonalsAlternating;
 my $path = Math::PlanePath::DiagonalsAlternating->new;
 my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

This path follows successive diagonals going from the Y axis down to the X axis and then back up again,

      5  |  16
         |   |\
      4  |  15  17
         |    \   \
      3  |   7  14  18
         |   |\   \   \
      2  |   6   8  13  19  ...
         |    \   \   \   \   \
      1  |   2   5   9  12  20  23
         |   |\   \   \   \   \   \
    Y=0  |   1   3-- 4  10--11  21--22
         +----------------------------
           X=0   1   2   3   4   5   6

The triangular numbers 1,3,6,10,etc k*(k+1)/2 are the start of each run up or down alternately on the X axis and Y axis. N=1,6,15,28,etc on the Y axis (Y even) are the hexagonal numbers j*(2j-1). N=3,10,21,36,etc on the X axis (X odd) are the hexagonal numbers of the second kind j*(2j+1).

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 pattern. For example to start at 0,

    n_start => 0
      4  |  14
      3  |   6 13
      2  |   5  7 12
      1  |   1  4  8 11
    Y=0  |   0  2  3  9 10
         +-----------------
           X=0  1  2  3  4

FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

"$path = Math::PlanePath::DiagonalsAlternating->new ()"
"$path = Math::PlanePath::DiagonalsAlternating->new (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 < 1" the return is an empty list, it being considered the path begins at 1.

FORMULAS

Rectangle to N Range

Within each row increasing X is increasing N, and in each column increasing Y is increasing N. So in a rectangle the lower left corner is the minimum N and the upper right is the maximum N.

    |               N max
    |     ----------+
    |    |  ^       |
    |    |  |       |
    |    |   ---->  |
    |    +----------
    |   N min
    +-------------------

OEIS

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

<http://oeis.org/A131179> (etc)

    n_start=1
      A131179    N on X axis (extra initial 0)
      A128918    N on Y axis (extra initial 1)
      A001844    N on X=Y diagonal
      A038722    permutation N at transpose Y,X
    n_start=0
      A319572    X coordinate
      A319573    Y coordinate
      A319571    X,Y coordinates together
      A003056    X+Y
      A004247    X*Y
      A049581    abs(X-Y)
      A048147    X^2+Y^2
      A004198    X bit-and Y
      A003986    X bit-or Y
      A003987    X bit-xor Y
      A004197    min(X,Y)
      A003984    max(X,Y)
      A101080    HammingDist(X,Y)
      A023531    dSum = dX+dY, being 1 at N=triangular+1 (and 0)
      A046092    N on X=Y diagonal
      A061579    permutation N at transpose Y,X
      A056011    permutation N at points by Diagonals,direction=up order
      A056023    permutation N at points by Diagonals,direction=down
         runs alternately up and down, both are self-inverse

The coordinates such as A003056 X+Y are the same here as in the Diagonals path. "DiagonalsAlternating" transposes X,Y -> Y,X in every second diagonal but forms such as X+Y are unchanged by swapping to Y+X.

SEE ALSO

Math::PlanePath, Math::PlanePath::Diagonals, Math::PlanePath::DiagonalsOctant

HOME PAGE

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

LICENSE

Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 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/>.

History