Math::PlanePath::MPeaks(3) points in expanding M shape

SYNOPSIS


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

DESCRIPTION

This path puts points in layers of an ``M'' shape 

         41                              49         7
         40  42                      48  50         6
         39  22  43              47  28  51         5
         38  21  23  44      46  27  29  52         4
         37  20   9  24  45  26  13  30  53         3
         36  19   8  10  25  12  14  31  54         2
         35  18   7   2  11   4  15  32  55         1
         34  17   6   1   3   5  16  33  56     <- Y=0
                          ^
         -4  -3  -2  -1  X=0  1   2   3   4

N=1 to N=5 is the first ``M'' shape, then N=6 to N=16 on top of that, etc. The centre goes half way down. Reckoning the N=1 to N=5 as layer d=1 then 

    Xleft = -d
    Xright = d
    Ypeak = 2*d - 1
    Ycentre = d - 1

Each ``M'' is 6 points longer than the preceding. The verticals are each 2 longer, and the centre diagonals each 1 longer. This step 6 is similar to the "HexSpiral".

The octagonal numbers N=1,8,21,40,65,etc k*(3k-2) are a straight line of slope 2 going up to the left. The octagonal numbers of the second kind N=5,16,33,56,etc k*(3k+2) are along the X axis to the right.

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
    40                              48
    39  41                      47  49
    38  21  42              46  27  50
    37  20  22  43      45  26  28  51
    36  19   8  23  44  25  12  29  52
    35  18   7   9  24  11  13  30  53
    34  17   6   1  10   3  14  31  54
    33  16   5   0   2   4  15  32  55

FUNCTIONS

See ``FUNCTIONS'' in Math::PlanePath for behaviour common to all path classes.
"$path = Math::PlanePath::MPeaks->new ()"
Create and return a new path object.
"($x,$y) = $path->n_to_xy ($n)"
Return the X,Y coordinates of point number $n on the path.

For "$n < 0.5" the return is an empty list, it being considered there are no negative points.

"$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 points as a squares of side 1, so the half-plane y>=-0.5 is entirely covered.

OEIS

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

<http://oeis.org/A045944> (etc) 

    n_start=1 (the default)
      A045944    N on X axis >= 1, extra initial 0
                   being octagonal numbers second kind
      A056106    N on Y axis, extra initial 1
      A056109    N on X negative axis <= -1
    n_start=0
      A049450    N on Y axis, extra initial 0, 2*pentagonal
    n_start=2
      A027599    N on Y axis, extra initial 6,2

HOME PAGE

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

LICENSE

Copyright 2011, 2012, 2013, 2014, 2015, 2016 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/>.

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