Math::PlanePath::HexSpiralSkewed -- integer points around a skewed hexagonal spiral
use Math::PlanePath::HexSpiralSkewed; my $path = Math::PlanePath::HexSpiralSkewed->new; my ($x, $y) = $path->n_to_xy (123);
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.
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
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.
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
.
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).
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.
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
Math::PlanePath, Math::PlanePath::HexSpiral, Math::PlanePath::HeptSpiralSkewed, Math::PlanePath::PentSpiralSkewed, Math::PlanePath::DiamondSpiral
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016 Kevin Ryde
This file is part of Math-PlanePath.
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