Kevin Ryde > Math-PlanePath > Math::PlanePath::WunderlichSerpentine

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NAME ^

Math::PlanePath::WunderlichSerpentine -- transpose parts of Peano curve, including coil order

SYNOPSIS ^

 use Math::PlanePath::WunderlichSerpentine;
 my $path = Math::PlanePath::WunderlichSerpentine->new (serpentine_type => '111_000_111');
 my ($x, $y) = $path->n_to_xy (123);

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

DESCRIPTION ^

This is an integer version of Walter Wunderlich's variations on the PeanoCurve. A "serpentine type" controls transposing of selected 3x3 sub-parts. The default is "alternating" 010,101,010 which transposes every second sub-part,

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

serpentine_type can be a string of 0s and 1s, with optional space, comma or _ separators at each group of 3,

    "011111011"         0/1 string
    "011,111,011"
    "011_111_011"
    "011 111 011"

or special values

    "alternating"       01010101.. the default
    "coil"              11111... all 1s described below
    "Peano"             00000... all 0s, gives PeanoCurve

Each "1" sub-part is transposed. The string is applied in order of the N parts, irrespective of what net reversals and transposes are in force on a particular part.

When no parts are transposed, which is a string of all 0s, the result is the same as the PeanoCurve. The special serpentine_type => "Peano" gives that.

Coil Order

serpentine_type => "coil" means "111 111 111" to transpose all parts. The result is like a coil viewed side-on,

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

           X=0   1   2   3   4   5   6   7   8

Whenever serpentine_type begins with a "1" the initial sub-part is transposed at each level. The first step N=0 to N=1 is kept fixed along the X axis, then the higher levels are transposed. For example in the coil above The N=9 to N=17 part is upwards, and then the N=81 to N=161 part is to the right, and so on.

Radix

The optional radix parameter gives the size of the sub-parts, similar to the PeanoCurve radix parameter (see "Radix" in Math::PlanePath::PeanoCurve). For example radix 5 gives

     radix => 5

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

Like the PeanoCurve if the radix is even then the ends of each sub-part don't join up. For example in radix 4 N=15 isn't next to N=16, nor N=31 to N=32, etc.

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

In the serpentine_type 0,1 form, any space, comma, etc, separators should group radix many values, so for example

    serpentine_type => "00000_11111_00000_00000_11111"

The intention is to do something friendly if the separators are not on such boundaries, so that say 000_111_000 can have a sensible meaning in a radix higher than 3. But exactly what is not settled, so always give a full string of desired 0,1 for now.

FUNCTIONS ^

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

$path = Math::PlanePath::WunderlichSerpentine->new ()
$path = Math::PlanePath::WunderlichSerpentine->new (serpentine_type => $str, 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. The radix should be an odd number, 3, 5, 7, 9 etc.

SEE ALSO ^

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

Walter Wunderlich "Uber Peano-Kurven", Elemente der Mathematik, 28(1):1-10, 1973. http://sodwana.uni-ak.ac.at/geom/mitarbeiter/wallner/wunderlich/ http://sodwana.uni-ak.ac.at/geom/mitarbeiter/wallner/wunderlich/pdf/125.pdf

HOME PAGE ^

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

LICENSE ^

Copyright 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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