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.
131211 ... 2
 \ \
14 43 10 23 1
  \ \ \
15 5 12 9 22 < Y=0
\ \  
16 678 21 1
\ 
17181920 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
131211 131211
 \ / \
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 (XY)/2, Y
skewed X,Y > plain 2*X+Y, Y
Corners
HexSpiralSkewed
is similar to the SquareSpiral
but cuts off the topright and bottomleft 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
2120191817 2
 \
22 8765 16 1
  \ \
23 9 1234 15 < Y=0
\ \ 
24 1011121314 ... 1
\ 
252627282930 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^22n+1
A056106 N on Y axis, 3n^2n+1
A056107 N on NorthWest 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 SouthEast 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 northeast
A033428 N on northwest diagonal, 3*k^2
A063436 N on southwest diagonal
A028896 N on southeast diagonal
SEE ALSO
Math::PlanePath, Math::PlanePath::HexSpiral, Math::PlanePath::HeptSpiralSkewed, Math::PlanePath::PentSpiralSkewed, Math::PlanePath::DiamondSpiral
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
Copyright 2010, 2011, 2012, 2013, 2014 Kevin Ryde
This file is part of MathPlanePath.
MathPlanePath 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.
MathPlanePath 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 MathPlanePath. If not, see <http://www.gnu.org/licenses/>.