/ 
Math-PlanePath-129
River stage one • 3 direct dependents • 4 total dependents
/ Math::PlanePath::HexSpiralSkewed

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, 2015, 2016, 2017, 2018, 2019, 2020 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/>.