Math::PlanePath::HexSpiral -- integer points around a hexagonal spiral
use Math::PlanePath::HexSpiral; my $path = Math::PlanePath::HexSpiral->new; my ($x, $y) = $path->n_to_xy (123);
This path makes a hexagonal spiral, with points spread out horizontally to fit on a square grid.
28 -- 27 -- 26 -- 25 3 / \ 29 13 -- 12 -- 11 24 2 / / \ \ 30 14 4 --- 3 10 23 1 / / / \ \ \ 31 15 5 1 --- 2 9 22 <- Y=0 \ \ \ / / 32 16 6 --- 7 --- 8 21 -1 \ \ / 33 17 -- 18 -- 19 -- 20 -2 \ 34 -- 35 ... -3 ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6
Each horizontal gap is 2, so for instance n=1 is at x=0,y=0 then n=2 is at x=2,y=0. The diagonals are just 1 across, so n=3 is at x=1,y=1. Each alternate row is offset from the one above or below. The result is a triangular lattice per "Triangular Lattice" in Math::PlanePath.
The octagonal numbers 8,21,40,65, etc 3*k^2-2*k fall on a horizontal straight line at y=-1. In general straight lines are 3*k^2 + b*k + c. The 3*k^2 goes diagonally up to the left, then b is a 1/6 turn counter-clockwise, or clockwise if negative. So b=1 goes horizontally to the left, b=2 diagonally down to the left, b=3 diagonally down to the right, etc.
An optional wider parameter makes the path wider, stretched along the top and bottom horizontals. For example
wider
$path = Math::PlanePath::HexSpiral->new (wider => 2);
gives
... 36----35 3 \ 21----20----19----18----17 34 2 / \ \ 22 8---- 7---- 6---- 5 16 33 1 / / \ \ \ 23 9 1---- 2---- 3---- 4 15 32 <- Y=0 \ \ / / 24 10----11----12----13----14 31 -1 \ / 25----26----27----28---29----30 -2 ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7
The centre horizontal from N=1 is extended by wider many extra places, then the path loops around that shape. The starting point N=1 is shifted to the left by wider many places to keep the spiral centred on the origin x=0,y=0. Each horizontal gap is still 2.
Each loop is still 6 longer than the previous, since the widening is basically a constant amount added into each loop.
See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.
$path = Math::PlanePath::HexSpiral->new ()
$path = Math::PlanePath::HexSpiral->new (wider => $w)
Create and return a new hex spiral object. An optional wider parameter widens the path, it defaults to 0 which is no widening.
($x,$y) = $path->n_to_xy ($n)
Return the X,Y coordinates of point number $n on the path.
$n
For $n < 1 the return is an empty list, it being considered the path starts at 1.
$n < 1
$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 $n in the path as a square of side 1.
$x,$y
$x
$y
Only every second square in the plane has an N, being those where X,Y both odd or both even. If $x,$y is a position without an N, ie. one of X,Y odd the other even, then the return is undef.
undef
Math::PlanePath, Math::PlanePath::HexSpiralSkewed, Math::PlanePath::HexArms, Math::PlanePath::TriangleSpiral, Math::PlanePath::TriangularHypot
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2010, 2011 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/>.
To install Math::PlanePath, copy and paste the appropriate command in to your terminal.
cpanm
cpanm Math::PlanePath
CPAN shell
perl -MCPAN -e shell install Math::PlanePath
For more information on module installation, please visit the detailed CPAN module installation guide.