NAME

Math::PlanePath::HexSpiral -- integer points around a hexagonal spiral

SYNOPSIS

use Math::PlanePath::HexSpiral;
my $path = Math::PlanePath::HexSpiral->new;
my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

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 resulting "triangles" between the points are flatter than they ought to be. Drawn on a square grid the angle up is 45 degrees making an isosceles right triangle instead of 60 for an equilateral triangle.

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.

Wider

An optional wider parameter makes the path wider, stretched along the top and bottom horizontals. For example

$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 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 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.

FUNCTIONS

$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.

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 $n in the path as a square of side 1.

Only every second square in the plane has an N. If $x,$y is a position without an N then the return is undef.

SEE ALSO

Math::PlanePath, Math::PlanePath::HexSpiralSkewed, Math::PlanePath::TriangleSpiral

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/>.