and 1 contributors

# NAME

Math::PlanePath::TerdragonMidpoint -- dragon curve midpoints

# SYNOPSIS

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

# DESCRIPTION

This is midpoints of an integer version of the terdragon curve by Davis and Knuth.

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

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

The points are the middle of each edge of a double-size `TerdragonCurve`.

``````                            ...
\
6             -----8-----      double size
\                TerdragonCurve
\               giving midpoints
5               7
\
\
4        -----6---- _
\         / \
\       /   \
3          5     4     3
\   /       \
\_/         \
2              _----2-----
\
\
1                1
\
\
Y=0 ->    +-----0-----.

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

For example in the `TerdragonCurve` N=3 to N=4 is X=3,Y=1 to X=2,Y=2 and that's doubled out here to X=6,Y=2 and X=4,Y=4 then the midpoint of those positions is X=5,Y=3 for N=3 in the `TerdragonMidpoint`.

The result is integer X,Y coordinates on every second point per "Triangular Lattice" in Math::PlanePath, but visiting only 3 of every 4 such triangular points, which in turn is 3 of 8 all integer X,Y points. The points used are a pattern of alternate rows with 1 of 2 points and 1 of 4 points. For example the Y=7 row is 1 of 2 and the Y=8 row is 1 of 4. Notice the pattern is the same when turned by 60 degrees.

``````    * * * * * * * * * * * * * * * * * * * *
*   *   *   *   *   *   *   *   *   *
* * * * * * * * * * * * * * * * * * * *
*   *   *   *   *   *   *   *   *
* * * * * * * * * * * * * * * * * * * *
*   *   *   *   *   *   *   *   *   *
* * * * * * * * * * * * * * * * * * * *
*   *   *   *   *   *   *   *   *
* * * * * * * * * * * * * * * * * * * *
*   *   *   *   *   *   *   *   *   *
* * * * * * * * * * * * * * * * * * * *
*   *   *   *   *   *   *   *   *
* * * * * * * * * * * * * * * * * * * *
*   *   *   *   *   *   *   *   *   *
* * * * * * * * * * * * * * * * * * * *``````

## Arms

Multiple copies of the curve can be selected, each advancing successively. Like the main `TerdragonCurve` the midpoint curve covers 1/6 of the plane and 6 arms rotated by 60, 120, 180, 240 and 300 degrees mesh together perfectly. With 6 arms all the alternating "1of2" and "1of4" points described above are visited.

`arms => 6` begins as follows. N=0,6,12,18,etc is the first arm (like the single curve 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                                 ...
/
...                           42
\                          /
43          19          36
\        /  \        /
37    25    13    30----24----18
\  /        \              /
31           7          12
\        /
20----14-----8-----2     1     6    35----41----47-..
\                          /        \
26           3     .     0          29
\        /                          \
..-44----38----32     9     4     5----11----17----23
/        \
15          10          34
/              \        /  \
21----27----33    16    28    40
/        \  /        \
39          22          46
/                          \
45                            ...
/
...``````

# FUNCTIONS

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

`\$path = Math::PlanePath::TerdragonMidpoint->new ()`

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.

`\$n = \$path->n_start()`

Return 0, the first N in the path.

## Level Methods

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

Return `(0, 3**\$level - 1)`, or for multiple arms return `(0, \$arms * 3**\$level - 1)`.

There are 3^level segments comprising the terdragon, or arms*3^level when multiple arms, numbered starting from 0.

# FORMULAS

## X,Y to N

An X,Y point can be turned into N by dividing out digits of a complex base b=w+1 where

``````    w = 1/2 + i * sqrt(3)/2            w^2     w
= 6th root of unity                 \   /
\ /
w^3=-1 -----o------ w^0=1
/ \
/   \
w^4     w^5``````

At each step the low ternary digit is formed from X,Y and an adjustment applied to move X,Y onto a multiple of w+1 ready to divide out w+1.

In the N points above it can be seen that each group of three N values make a straight line, such as N=0,1,2, or N=3,4,5 etc. The adjustment moves the two ends N=0mod3 or N=2mod3 to the centre N=1mod3. The centre N=1mod3 position is always a multiple of w+1.

The angles and positions for the N triples follow a 12-point pattern as follows, where each / \ or - is a point on the path (any arm).

``````     \   /   /   \   /   /   \   /   /   \   /   /   \
- \ / \ - - - \ / \ - - - \ / \ - - - \ / \ - - -
/   \   /   /   \   /   /   \   /   /   \   /
\ - - - \ / \ - - - \ / \ - - - \ / \ - - - \ / \
\   /   /   \   /   /   \   /   /   \   /   /   \
- \ / \ - - - \ / \ - - - \ / \ - - - \ / \ - - -
/   \   /   /   \   /   /   \   /   /   \   /
\ - - - \ / \ - - - \ / \ - - - \ / \ - - - \ / \
\   /   /   \   /   /   \   /   /   \   /   /   \
- \ / \ - - - \ / \ - - - \ / \ - - - \ / \ - - -
/   \   /   /   \   /   /   \   /   /   \   /
\ - - - \ / \ - - - \ / \ - - - \ / \ - - - \ / \
\   /   /   \   /   /   \   /   /   \   /   /   \
- \ / \ - - - \ / \ - - - \ / \ - - - \ / \ - - -
/   \   /   /   \   /   /   \   /   /   \   /
\ - - - \ / \ - - - \ / \ - - - \ / \ - - - \ / \
\   /   /   \   /   /   \   /   /   \   /   /   \
- \ / \ - - - \ / \ - - - \ / \ - - - \ / \ - - -
/   \   /   /   \   /   /   \   /   /   \   /
\ - - - \ / \ - - - \ / \ - - - \ / \ - - - \ / \
\   /   /   \   /   /   \   /   /   \   /   /   \
- \ / \ - - - \ / \ - - - \ / \ - - - \ / \ - - -
/   \   /   /   \   /   /   \   /   /   \   /
\ - - - \ / \ - - - \ / \ - - - \ / \ - - - \ / \``````

In the current code a 12x12 table is used, indexed by X mod 12 and Y mod 12. With Xadj and Yadj from there

``````    Ndigit = (X + 1) mod 3      # N digits low to high

Xm = X + Xadj[X mod 12, Y mod 12]
Ym = Y + Yadj[X mod 12, Y mod 12]

new X,Y = (Xm,Ym) / (w+1)
= (Xm,Ym) * (2-w) / 3
= ((Xm+Ym)/2, (Ym-(Xm/3))/2)``````

Is there a good aX+bY mod 12 or mod 24 for a smaller table? Maybe X+3Y like the digit? Taking C=(X-Y)/2 in triangular coordinate style can reduce the table to 6x6.

Points not reached by the curve (ie. not the 3 of 4 triangular or 3 of 8 rectangular described above) can be detected with `undef` or suitably tagged entries in the adjustment table.

The X,Y reduction stops at the midpoint of the first triple of the originating arm. So X=3,Y=1 which is N=1 for the first arm, and that point rotated by 60,120,180,240,300 degrees for the others. If only some of the arms are of interest then reaching one of the others means the original X,Y was outside the desired region.

``````    Arm     X,Y Endpoint
---     ------------
0        3,1
1        0,2
2       -3,1
3       -3,-1
4        0,-2
5        3,-1``````

For the odd arms 1,3,5 each digit of N must be flipped 2-digit so 0,1,2 becomes 2,1,0,

``````    if arm odd
then  N = 3**numdigits - 1 - N``````

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