NAME

Math::PlanePath::DragonRounded -- dragon curve with rounded corners

SYNOPSIS

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

DESCRIPTION

This is a version of the dragon curve by Heighway, Harter, et al, done with two points per edge and skipping vertices so as to make rounded-off corners,

                          17-16              9--8                 6
                         /     \           /     \
                       18       15       10        7              5
                        |        |        |        |
                       19       14       11        6              4
                         \        \     /           \
                          20-21    13-12              5--4        3
                               \                          \
                                22                          3     2
                                 |                          |
                                23                          2     1
                               /                          /
        33-32             25-24                    .  0--1       Y=0
       /     \           /
     34       31       26                                        -1
      |        |        |
     35       30       27                                        -2
       \        \     /
        36-37    29-28    44-45                                  -3
             \           /     \
              38       43       46                               -4
               |        |        |
              39       42       47                               -5
                \     /        /
                 40-41    49-48                                  -6
                         /
                       50                                        -7
                        |
                       ...


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

The two points on an edge have one of X or Y a multiple of 3, and the other Y or X at 1 mod 3 or 2 mod 3. For example the N=19 and N=20 are on the X=-9 edge (a multiple of 3), and at Y=4 and Y=5 (1 and 2 mod 3).

The "rounding" of the corners ensures that for example N=13 and N=21 don't touch as they approach X=-6,Y=3. The curve never crosses itself.

Arms

The dragon curve fills a quarter of the plane and four copies mesh together rotated by 90, 180 and 270 degrees. The arms parameter can choose 1 to 4 curve arms, successively advancing. For example arms => 4 gives

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



     ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^
    -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6

With 4 arms all 3x3 blocks are visited, using 4 out of 9 points in each.

FUNCTIONS

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

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

$path = Math::PlanePath::DragonRounded->new (arms => $aa)

Create and return a new path object.

The optional arms parameter makes a multi-arm curve. The default is 1 for just one arm.

($x,$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.

$n = $path->n_start()

Return 0, the first N in the path.

SEE ALSO

Math::PlanePath, Math::PlanePath::DragonCurve, Math::PlanePath::DragonMidpoint

HOME PAGE

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

LICENSE

Copyright 2011 Kevin Ryde

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/>.