NAME¶
Math::PlanePath::MPeaks -- 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
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
SEE ALSO¶
Math::PlanePath, Math::PlanePath::PyramidSides
HOME PAGE¶
<
http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE¶
Copyright 2011, 2012, 2013, 2014 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/>.