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
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 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/>.