Provided by: libmath-planepath-perl_125-1_all 
NAME
Math::PlanePath::HexSpiralSkewed -- integer points around a skewed hexagonal spiral
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
SEE ALSO
Math::PlanePath, Math::PlanePath::HexSpiral, Math::PlanePath::HeptSpiralSkewed,
Math::PlanePath::PentSpiralSkewed, Math::PlanePath::DiamondSpiral
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017 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/>.