++ed by:
Kevin Ryde
and 1 contributors

# NAME

Math::PlanePath::TerdragonRounded -- triangular dragon curve, with rounded corners

# SYNOPSIS

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

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

# DESCRIPTION

This is a version of the terdragon curve with rounded-off corners,

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

^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^
-8 -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7  8``````

The plain `TerdragonCurve` is tripled in size and two points on each 3-long edge are visited by the `TerdragonRounded` here.

## Arms

Multiple copies of the curve can be selected, each advancing successively. The curve is 1/6 of the plane (like the plain terdragon) and 6 arms rotated by 60, 120, 180, 240 and 300 degrees mesh together perfectly.

`arms => 6` begins as follows. N=0,6,12,18,etc is the first arm (the curve shown above), then N=1,7,13,19 the second copy rotated 60 degrees, N=2,8,14,20 the third rotated 120, etc.

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

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

# FUNCTIONS

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

`\$path = Math::PlanePath::TerdragonRounded->new ()`
`\$path = Math::PlanePath::TerdragonRounded->new (arms => \$count)`

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.

Fractional positions give an X,Y position along a straight line between the integer positions.

# FORMULAS

## X,Y Visited

When arms=6 all "hex centred" points of the plane are visited, being those points with

``    X+3Y mod 6 == 2 or 4        "hex_centred"``

Jorg Arndt `http://www.jjj.de/fxt/#fxtbook` section 1.31.4 "Terdragon and Hexdragon", where this rounded terdragon is called hexdragon.

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