NAME

Math::PlanePath::TriangleSpiralSkewed -- integer points drawn around a skewed equilateral triangle

SYNOPSIS

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

DESCRIPTION

This path makes an spiral shaped as an equilateral triangle (each side the same length), but skewed to the left to fit on a square grid,

    16                              4
     |\
    17 15                           3
     |   \
    18  4 14                        2
     |  |\  \
    19  5  3 13                     1
     |  |   \  \
    20  6  1--2 12 ...         <- Y=0
     |  |         \  \
    21  7--8--9-10-11 30           -1
     |                  \
    22-23-24-25-26-27-28-29        -2

           ^
    -2 -1 X=0 1  2  3  4  5

The properties are the same as the spread-out TriangleSpiral. The triangle numbers fall on straight lines as the do in the TriangleSpiral but the skew means the top corner goes up at an angle to the vertical and the left and right downwards are different angles plotted (but are symmetric by N count).

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,

    15        n_start => 0
     |\
    16 14
     |   \
    17  3 13 ...
     |  |\  \  \
    18  4  2 12 31
     |  |   \  \  \
    19  5  0--1 11 30
     |  |         \  \
    20  6--7--8--9-10 29
     |                  \
    21-22-23-24-25-26-27-28

With this adjustment for example the X axis N=0,1,11,30,etc is (9k-7)*k/2, the hendecagonal numbers (11-polygonals). And N=0,8,25,etc is the hendecagonals of the second kind, (9k-7)*k/2 for k negative.

FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

$path = Math::PlanePath::TriangleSpiralSkewed->new ()

$path = Math::PlanePath::TriangleSpiralSkewed->new (n_start => $n)

Create and return a new skewed triangle spiral object.

($x,$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 centred in a square of side 1, so the entire plane is covered.

OEIS

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

    http://oeis.org/A117625  (etc)

    n_start=1 (default)
      A010054     turn 1=left,0=straight, extra initial 1

      A117625     N on X axis
      A064226     N on Y axis, but without initial value=1
      A006137     N on X negative
      A064225     N on Y negative
      A081589     N on X=Y leading diagonal
      A038764     N on X=Y negative South-West diagonal
      A081267     N on X=-Y negative South-East diagonal
      A060544     N on ESE slope dX=+2,dY=-1
      A081272     N on SSE slope dX=+1,dY=-2

      A217010     permutation N values of points by SquareSpiral order
      A217291      inverse

    n_start=0
      A051682     N on X axis (11-gonal numbers)
      A081268     N on X=1 vertical (next to Y axis)
      A062725     N on Y negative
      A081275     N on X=Y+1 North-East diagonal
      A062728     N on South-East diagonal (11-gonal second kind)
      A081266     N on X=Y negative South-West diagonal
      A081270     N on X=1-Y North-West diagonal, starting N=3
      A081271     N on dX=-1,dY=2 NNW slope up from N=1 at X=1,Y=0

SEE ALSO

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

HOME PAGE

http://user42.tuxfamily.org/math-planepath/index.html

LICENSE

Copyright 2010, 2011, 2012, 2013 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/>.