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Math-PlanePath-129
River stage one • 3 direct dependents • 4 total dependents
/ Math::PlanePath::CubicBase

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.

Level Methods

($n_lo, $n_hi) = $path->level_to_n_range($level)

Return (0, $radix**$level - 1).

SEE ALSO

Math::PlanePath, Math::PlanePath::ImaginaryBase, Math::PlanePath::ImaginaryHalf

HOME PAGE

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

LICENSE

Copyright 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 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/>.