NAME

Math::PlanePath::MathImageAR2W2Curve -- 2x2 self-similar of four pattern parts

SYNOPSIS

use Math::PlanePath::MathImageAR2W2Curve;
my $path = Math::PlanePath::MathImageAR2W2Curve->new;
my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

In progress ...

This is an integer version of the AR2W2 curve by Asano, Ranjan, Roos, Welzl and Widmayer.

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

It makes a 2x2 expanding pattern with a mixture of "U" parts in the style of HilbertCurve and "Z" parts in the style of ZOrderCurve. The mixture is designed to improve some kinds of locality measures.

There's four base patterns A to D, plus mirroring,

A1    C1---B2        A2   C2   D1
         \                 | \  |
      D2---B1             B1   B2

B1    C2---B1        B2   B2---C1
       |    |              |    |
      D1   B2             B1   D2

C1    B1---A1        C2   A2---B2
       |    |              |    |
      A2   B2             B1   A1

D1    A2---C2        D2   C1---A1
       |    |              |    |
      D1   A2             A1   D2

The starting pattern is the A1 "Z" shape and above that D2. Notice the starting sub-part of D2 is A1 and in turn the starting sub-part of A1 is D2, so those two alternate at progressively higher levels. The sub-parts of the two then reach all of the others, in transpose and straight forms, forward and reverse, and rotated four directions.

FUNCTIONS

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

$path = Math::PlanePath::MathImageAR2W2Curve->new ()

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.

SEE ALSO

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

Asano, Ranjan, Roos, Welzl and Widmayer "Space-Filling Curves and Their Use in the Design of Geometric Data Structures", Theoretical Computer Science, 181(1):3-15, 1997. And LATIN'95 Theoretical Informatics on Google Books books.google.com.au/books?isbn=3540591753.