Kevin Ryde > Math-PlanePath > Math::PlanePath::GreekKeySpiral

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NAME ^

Math::PlanePath::GreekKeySpiral -- square spiral with Greek key motif

SYNOPSIS ^

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

DESCRIPTION ^

This path makes a spiral with a Greek key scroll motif,

    39--38--37--36  29--28--27  24--23                      5
     |           |   |       |   |   |
    40  43--44  35  30--31  26--25  22                      4
     |   |   |   |       |           |
    41--42  45  34--33--32  19--20--21  ...                 3
             |               |           |
    48--47--46   5---6---7  18  15--14  99  96--95          2
     |           |       |   |   |   |   |   |   |
    49  52--53   4---3   8  17--16  13  98--97  94          1
     |   |   |       |   |           |           |
    50--51  54   1---2   9--10--11--12  91--92--93     <- Y=0
             |                           |
    57--56--55  68--69--70  77--78--79  90  87--86         -1
     |           |       |   |       |   |   |   |
    58  61--62  67--66  71  76--75  80  89--88  85         -2
     |   |   |       |   |       |   |           |
    59--60  63--64--65  72--73--74  81--82--83--84         -3

                 ^

    -3  -2  -1  X=0  1   2   3   4   5   6   7   8 ...

The repeating figure is a 3x3 pattern

       |
       *   *---*
       |   |   |      right vertical
       *---*   *      going upwards
               |
       *---*---*
       |

The turn excursion is to the outside of the 3-wide channel and forward in the direction of the spiral. The overall spiraling is the same as the SquareSpiral, but composed of 3x3 sub-parts.

Sub-Part Joining

The verticals have the "entry" to each figure on the inside edge, as for example N=90 to N=91 above. The horizontals instead have it on the outside edge, such as N=63 to N=64 along the bottom. The innermost N=1 to N=9 is a bottom horizontal going right.

      *---*---*
      |       |        bottom horizontal
      *---*   *        going rightwards
          |   |
    --*---*   *-->

On the horizontals the excursion part is still "forward on the outside", as for example N=73 through N=76, but the shape is offset. The way the entry is alternately on the inside and outside for the vertical and horizontal is necessary to make the corners join.

Turn

An optional turns => $integer parameter controls the turns within the repeating figure. The default is turns=>2. Or for example turns=>4 begins

    turns => 4

    105-104-103-102-101-100  79--78--77--76--75  62--61--60--59
      |                   |   |               |   |           |
    106 119-120-121-122  99  80  87--88--89  74  63  66--67  58
      |   |           |   |   |   |       |   |   |   |   |   |
    107 118 115-114 123  98  81  86--85  90  73  64--65  68  57
      |   |   |   |   |   |   |       |   |   |           |   |
    108 117-116 113 124  97  82--83--84  91  72--71--70--69  56
      |           |   |   |               |                   |
    109-110-111-112 125  96--95--94--93--92  51--52--53--54--55
                      |                       |
    130-129-128-127-126  17--18--19--20--21  50  37--36--35--34
      |                   |               |   |   |           |
    131 144-145-146-147  16   9-- 8-- 7  22  49  38  41--42  33
      |   |           |   |   |       |   |   |   |   |   |   |
    132 143 140-139 148  15  10--11   6  23  48  39--40  43  32
      |   |   |   |   |   |       |   |   |   |           |   |
    133 142-141 138 149  14--13--12   5  24  47--46--45--44  31
      |           |   |               |   |                   |
    134-135-136-137 150   1-- 2-- 3-- 4  25--26--27--28--29--30
                      |
             ..-152-151

The count of turns is chosen to make turns=>0 a straight line, the same as the SquareSpiral. turns=>1 is a single wiggle,

    turns => 1

    66--65--64  61--60  57--56  53--52--51
     |       |   |   |   |   |   |       |
    67--68  63--62  59--58  55--54  49--50
         |                           |    
    70--69  18--17--16  13--12--11  48--47
     |       |       |   |       |       |
    71--72  19--20  15--14   9--10  45--46
         |       |           |       |    
       ...  22--21   2-- 3   8-- 7  44--43
             |       |   |       |       |
            23--24   1   4-- 5-- 6  41--42
                 |                   |    
            26--25  30--31  34--35  40--39
             |       |   |   |   |       |
            27--28--29  32--33  36--37--38

In general the repeating figure is a square of turns+1 points on each side, spiralling in and then out again.

FUNCTIONS ^

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

$path = Math::PlanePath::GreekKeySpiral->new ()
$path = Math::PlanePath::GreekKeySpiral->new (turns => $integer)

Create and return a new Greek key spiral object. The default turns is 2.

($x,$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.

SEE ALSO ^

Math::PlanePath, Math::PlanePath::SquareSpiral

Jo Edkins Greek Key pages http://gwydir.demon.co.uk/jo/greekkey/index.htm

HOME PAGE ^

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

LICENSE ^

Copyright 2010, 2011, 2012, 2013, 2014 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/>.

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