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 sequence is the same as the plain HexSpiral, but this arrangement fits more points on a square grid. The skew pushes the top horizontal to the left, as illustrated by the following parts of the two. 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

The kinds of 3*k^2 number sequences which fall on straight lines in the plain HexSpiral also fall on straight lines when skewed. See Math::PlanePath::HexSpiral for notes on this.

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 1 to 2 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 that plain HexSpiral.

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.

FUNCTIONS

$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.

($x,$y) = $path->n_to_xy ($n)

Return the x,y coordinates of point number $n on the path.

For $n < 1 the return is an empty list, it being considered the path starts at 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 point in the path as a square of side 1.

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

Math-PlanePath is Copyright 2010, 2011 Kevin Ryde

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/>.