NAME

Math::PlanePath::MathImagePeanoRounded -- 3x3 self-similar quadrant traversal

SYNOPSIS

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

 # or another radix digits ...
 my $path5 = Math::PlanePath::MathImagePeanoRounded->new (radix => 5);

DESCRIPTION

This is a version of the PeanoCurve with rounded-off corners,

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

Radix

The radix parameter can do the calculation in a base other than 3, using the same kind of direction reversals. For example radix 5 gives 5x5 groups,

      9  |     41-42       45-46       49-...
         |    /     \     /     \     /   
      8  |  40       43-44       47-48    
         |   |
      7  |  39       36-35       32-31    
         |    \     /     \     /     \   
      6  |     38-37       34-33       30 
         |                              | 
      5  |     21-22       25-26       29 
         |    /     \     /     \     /   
      4  |  20       23-24       27-28    
         |   |                            
      3  |  19       16-15       12-11    
         |    \     /     \     /     \   
      2  |     18-17       14-13       10 
         |                              | 
      1  |      1--2        5--6        9 
         |    /     \     /     \     /   
     Y=0 |   0        3--4        7--8    
         |
         +---------------------------------
           X=0   1   2   3   4   5   6   7 

If the radix is even then the ends of each group don't join up. For example in radix 4 N=31 isn't next to N=32, etc.

         |       30   29             26   25        32 
      3  |  31             28   27             24       33
         |       17   18             21   22       
      2  |  16             19   20             23  
         |       14   13             10    9       
      1  |  15             12   11              8  
         |        1    2              5    6       
     Y=0 |   0              3    4              7      
         +------------------------------------------------
           X=0   1   2   4   5   6   7   8   9  10  11  12

Even sizes can be made to join using other patterns, but this module is just Peano's digit construction. For 2x2 groupings see HilbertCurve (which is essentially the only way to join up in 2x2). For bigger groupings there's various ways.

FUNCTIONS

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

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

$path = Math::PlanePath::MathImagePeanoRounded->new (radix => $r)

Create and return a new path object.

The optional radix parameter gives the base for digit splitting. The default is ternary, radix => 3.

($x,$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.

SEE ALSO

Math::PlanePath, Math::PlanePath::PeanoCurve, Math::PlanePath::DragonRounded

Guiseppe Peano, "Sur une courbe, qui remplit toute une aire plane", Mathematische Annalen, volume 36, number 1, 1890, p157-160

    http://www.springerlink.com/content/w232301n53960133/
    DOI 10.1007/BF01199438