Provided by: libmath-planepath-perl_113-1_all bug

NAME

       Math::PlanePath::CubicBase -- replications in three directions

SYNOPSIS

        use Math::PlanePath::CubicBase;
        my $path = Math::PlanePath::CubicBase->new (radix => 4);
        my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

       This is a pattern of replications in three directions 0, 120 and 240 degrees.

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

                                              ^
           -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6

       The points are on a triangular grid by using every second integer X,Y, as per "Triangular
       Lattice" in Math::PlanePath.  All points on that triangular grid are visited.

       The initial N=0,N=1 is replicated at +120 degrees.  Then that trapezoid at +240 degrees

           +-----------+                       +-----------+
            \  2     3  \                       \  2     3  \
             +-----------+                       \           \
               \  0     1  \                       \  0     1  \
                +-----------+             ---------  -----------+
                                          \  6     7  \
             replicate +120deg              \          \    rep +240deg
                                             \  4     5 \
                                              +----------+

       Then that bow-tie N=0to7 is replicated at 0 degrees again.  Each replication is 1/3 of the
       circle around, 0, 120, 240 degrees repeating.  The relative layout within a replication is
       unchanged.

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

       The radial distance doubles on every second replication, so N=1 and N=2 are at 1 unit from
       the origin, then N=4 and N=8 at 2 units, then N=16 and N=32 at 4 units.  N=64 is not shown
       but is then at 8 units away (X=8,Y=0).

       This is similar to the "ImaginaryBase", but where that path repeats in 4 directions based
       on i=squareroot(-1), here it's 3 directions based on w=cuberoot(1) = -1/2+i*sqrt(3)/2.

   Radix
       The "radix" parameter controls the "r" used to break N into X,Y.  For example radix 4
       gives 4x4 blocks, with r-1 replications of the preceding level at each stage.

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

                                                          ^
           -15-14-13-12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6

       Notice the parts always replicate away from the origin, so the block N=16 to N=31 is near
       the origin at X=-4, then N=32,48,64 are further away.

       In this layout the replications still mesh together perfectly and all points on the
       triangular grid are visited.

FUNCTIONS

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

       "$path = Math::PlanePath::CubicBase->new ()"
       "$path = Math::PlanePath::CubicBase->new (radix => $r)"
           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::ImaginaryBase, Math::PlanePath::ImaginaryHalf

HOME PAGE

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

LICENSE

       Copyright 2012, 2013 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/>.