SYNOPSIS
use Math::PlanePath::HexSpiralSkewed;
my $path = Math::PlanePath::HexSpiralSkewed->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This path makes a hexagonal spiral with points skewed so as to fit a square grid and fully cover the plane.
13--12--11 ... 2 | \ \ 14 4---3 10 23 1 | | \ \ \ 15 5 1---2 9 22 <- Y=0 \ \ | | 16 6---7---8 21 -1 \ | 17--18--19--20 -2 ^ ^ ^ ^ ^ ^ -2 -1 X=0 1 2 3 ...
The kinds of N=3*k^2 numbers which fall on straight lines in the plain "HexSpiral" also fall on straight lines when skewed. See Math::PlanePath::HexSpiral for notes on this.
Skew
The skewed path is the same shape as the plain "HexSpiral", but fits more points on a square grid. The skew pushes the top horizontal to the left, as shown by the following parts, and the bottom horizontal is similarly skewed but to the right.
HexSpiralSkewed HexSpiral 13--12--11 13--12--11 | \ / \ 14 10 14 10 | \ / \ 15 9 15 9 -2 -1 X=0 1 2 -4 -3 -2 X=0 2 3 4
In general the coordinates can be converted each way by
plain X,Y -> skewed (X-Y)/2, Y skewed X,Y -> plain 2*X+Y, Y
Corners
"HexSpiralSkewed" is similar to the "SquareSpiral" but cuts off the top-right and bottom-left corners so that each loop is 6 steps longer than the previous, whereas for the "SquareSpiral" it's 8. See ``Corners'' in Math::PlanePath::SquareSpiral for other corner cutting.Wider
An optional "wider" parameter makes the path wider, stretched along the top and bottom horizontals. For example
$path = Math::PlanePath::HexSpiralSkewed->new (wider => 2);
gives
21--20--19--18--17 2 | \ 22 8---7---6---5 16 1 | | \ \ 23 9 1---2---3---4 15 <- Y=0 \ \ | 24 10--11--12--13--14 ... -1 \ | 25--26--27--28--29--30 -2 ^ ^ ^ ^ ^ ^ ^ ^ -4 -3 -2 -1 X=0 1 2 3 ...
The centre horizontal from N=1 is extended by "wider" many further places, then the path loops around that shape. The starting point 1 is shifted to the left by wider/2 places (rounded up to an integer) to keep the spiral centred on the origin X=0,Y=0.
Each loop is still 6 longer than the previous, since the widening is basically a constant amount added into each loop. The result is the same as the plain "HexSpiral" of the same widening too. The effect looks better in the plain "HexSpiral".
N Start
The default is to number points starting N=1 as shown above. An optional "n_start" can give a different start with the same shape etc. For example to start at 0,
n_start => 0 27 26 25 24 3 28 12 11 10 23 2 29 13 3 2 9 22 1 30 14 4 0 1 8 21 ... <- Y=0 31 15 5 6 7 20 39 -1 32 16 17 18 19 38 -2 33 34 35 36 37 -3 -3 -2 -1 X=0 1 2 3 4
In this numbering the X axis N=0,1,8,21,etc is the octagonal numbers 3*X*(X+1).
FUNCTIONS
See ``FUNCTIONS'' in Math::PlanePath for behaviour common to all path classes.- "$path = Math::PlanePath::HexSpiralSkewed->new ()"
- "$path = Math::PlanePath::HexSpiralSkewed->new (wider => $w)"
- Create and return a new hexagon spiral object. An optional "wider" parameter widens the spiral path, it defaults to 0 which is no widening.
- "$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 in the path as a square of side 1.
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
- <http://oeis.org/A056105> (etc)
A056105 N on X axis, 3n^2-2n+1 A056106 N on Y axis, 3n^2-n+1 A056107 N on North-West diagonal, 3n^2+1 A056108 N on X negative axis, 3n^2+n+1 A056109 N on Y negative axis, 3n^2+2n+1 A003215 N on South-East diagonal, centred hexagonals n_start=0 A000567 N on X axis, octagonal numbers A049450 N on Y axis A049451 N on X negative axis A045944 N on Y negative axis, octagonal numbers second kind A062783 N on X=Y diagonal north-east A033428 N on north-west diagonal, 3*k^2 A063436 N on south-west diagonal A028896 N on south-east diagonal
LICENSE
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016 Kevin RydeThis 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/>.