Kevin Ryde > Math-PlanePath > Math::PlanePath::SierpinskiCurveStair

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Module Version: 116   Source  

NAME ^

Math::PlanePath::SierpinskiCurveStair -- Sierpinski curve with stair-step diagonals

SYNOPSIS ^

 use Math::PlanePath::SierpinskiCurveStair;
 my $path = Math::PlanePath::SierpinskiCurveStair->new (arms => 2);
 my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION ^

This is a variation on the SierpinskiCurve with stair-step diagonal parts.

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

The tiling is the same as the SierpinskiCurve, but each diagonal is a stair step horizontal and vertical. The correspondence is

    SierpinskiCurve        SierpinskiCurveStair

                7--                   12--
              /                        |
             6                     10-11
             |                      |
             5                      9--8
              \                        |
       1--2     4             2--3  6--7
     /     \  /               |  |  |
    0        3             0--1  4--5

So the SierpinskiCurve N=0 to N=1 diagonal corresponds to N=0 to N=2 here, and N=2 to N=3 corresponds to N=3 to N=5. The join section N=3 to N=4 gets an extra point at N=6 here, and later similar N=19, etc.

Diagonal Length

The diagonal_length option can make longer diagonals, still in stair-step style. For example diagonal_length => 4,

    10  |                                 36-37
        |                                  |  |
     9  |                              34-35 38-39
        |                               |        |
     8  |                           32-33       40-41
        |                            |              |
     7  |                        30-31             42-43
        |                         |                    |
     6  |                     28-29                   44-45
        |                      |                          |
     5  |                     27-26                   47-46
        |                         |                    |
     4  |                8--9    25-24             49-48    ...
        |                |  |        |              |        |
     3  |             6--7 10-11    23-22       51-50    62-63
        |             |        |        |        |        |
     2  |          4--5       12-13    21-20 53-52    60-61
        |          |              |        |  |        |
     1  |       2--3             14-15 18-19 54-55 58-59
        |       |                    |  |        |  |
    Y=0 |    0--1                   16-17       56-57
        |
        +------------------------------------------------------
          ^
         X=0 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17

The length is reckoned from N=0 to the end of the first side N=8, which is X=1 to X=5 for length 4 units.

Arms

The optional arms parameter can give up to eight copies of the curve, each advancing successively. For example arms => 8,

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

The multiplies of 8 (or however many arms) N=0,8,16,etc is the original curve, and the further mod 8 parts are the copies.

The middle "." shown is the origin X=0,Y=0. It would be more symmetrical to have the origin the middle of the eight arms, which would be X=-0.5,Y=-0.5 in the above, but that would give fractional X,Y values. Apply an offset X+0.5,Y+0.5 to centre if desired.

Level Ranges

For diagonal_length = L and reckoning the first diagonal side N=0 to N=2L as level 0, a level extends out to a triangle

    Nlevel = ((6L+4)*4^level - 4) / 3
    Xlevel = (L+2)*2^level - 1

For example level 2 in the default L=1 goes to N=((6*1+4)*4^2-4)/3=52 and Xlevel=(1+2)*2^2-1=11. Or in the L=4 sample above level 1 is N=((6*4+4)*4^1-4)/3=36 and Xlevel=(4+2)*2^1-1=11.

The power-of-4 in Nlevel is per the plain SierpinskiCurve, with factor 2L+1 for the points making the diagonal stair. The "/3" arises from the extra points between replications. They become a power-of-4 series

    Nextras = 1+4+4^2+...+4^(level-1) = (4^level-1)/3

For example level 1 is Nextras=(4^1-1)/3=1, being point N=6 in the default L=1. Or for level 2 Nextras=(4^2-1)/3=5 at N=6 and N=19,26,33,46.

The curve doesn't visit all the points in the eighth of the plane below the X=Y diagonal. In general Nlevel+1 many points of the triangular area Xlevel*(Xlevel-1)/2 are visited, for a filled fraction which approaches a constant

    FillFrac = Nlevel / (Xlevel*(Xlevel-1)/2)
            -> 4/3 * (3L+2)/(L+2)^2

For example the default L=1 has FillFrac=20/27=0.74. Or L=2 FillFrac=2/3=0.66. As the diagonal length increases the fraction decreases due to the growing holes in the pattern.

FUNCTIONS ^

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

$path = Math::PlanePath::SierpinskiCurveStair->new ()
$path = Math::PlanePath::SierpinskiCurveStair->new (diagonal_length => $L, arms => $A)

Create and return a new path object.

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

Return the X,Y coordinates of point number $n on the path. Points begin at 0 and if $n < 0 then the return is an empty list.

Fractional positions give an X,Y position along a straight line between the integer positions.

$n = $path->n_start()

Return 0, the first N in the path.

SEE ALSO ^

Math::PlanePath, Math::PlanePath::SierpinskiCurve

HOME PAGE ^

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

LICENSE ^

Copyright 2011, 2012, 2013, 2014 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/>.

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