NAME
Math::PlanePath::MathImageTerdragonCurve -- triangular dragon curve
SYNOPSIS
use Math::PlanePath::MathImageTerdragonCurve;
my $path = Math::PlanePath::MathImageTerdragonCurve->new;
my ($x, $y) = $path->n_to_xy (123);DESCRIPTION
In progress ...
This is the terdragon curve by Davis and Knuth,
30 28 7
/ \ / \
31/34 -------- 26/29/32 --------- 27 6
\ / \
24/33/42 ---------- 22/25 5
/ \ / \
40/43/46 -------- 20/23/44 -------- 12/21 10 4
\ / \ / \ / \
18/45 --------- 13/16/19 ------ 8/11/14 -------- 9 3
\ / \ / \
17 6/15 --------- 4/7 2
\ / \
2/5 --------- 3 1
\
0 ----------- 1 <- Y=0
^ ^ ^ ^ ^ ^ ^ ^
-4 -3 -2 -1 X=0 1 2 3The curve visits "inner" X,Y points three times, and outer points either once or twice. The first triple point is X=1,Y=3 which is N=8, N=11 and N=14. The segments N=7,8,9 make a vertex there, as does N=10,11,12 and N=13,14,15. they touch, but the path doesn't cross itself. The tripled vertices are all like this, touching but not crossing, and no edges repeat.
The first step N=1 is to the right along the X axis and the path then slowly spirals counter-clockwise and progressively fatter. The end of each replication is
Nlevel = 3^leveland that point is at level*30 degrees around,
Nlevel X,Y angle (degrees)
------ ----- -----
1 1,0 0
3 3,1 30
9 3,3 60
27 0,6 90
81 -9,9 120
243 -27,9 150
729 -54,0 180The following is points N=0 to N=3^6=729 with the N=0 origin marked "o" and the N=729 end marked "+". It's gone half-circle around to 180 degrees,
* * * *
* * * * * * * *
* * * * * * * *
* * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * *
* + * * * * * * * * * * * * * * * * o *
* * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * *
* * * * * * * *
* * * * * * * *
* * * *Turns
At each point N the curve always turns 120 degrees either to the left or right, it never goes straight ahead. If N is written in ternary then ...
Ndigit Turn
------ ----
0
1
2Arms
The curve fills a quarter of the plane and six copies mesh together perfectly when rotated by 60, 120, 180, 240 and 300 degrees. The arms parameter can choose 1 to 6 curve arms, successively advancing.
For example arms => 6 begins as follows, with N=0,6,12,18,etc being one arm, N=1,7,13,19 the second, N=2,8,14,20 the third, etc.
17 --- 13/6 --- 0/1/2/3 --- 4/15 --- 19With four arms every X,Y point is visited twice (except the origin 0,0 where all four begin) and every edge between the points is traversed once.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.
$path = Math::PlanePath::MathImageTerdragonCurve->new ()$path = Math::PlanePath::MathImageTerdragonCurve->new (arms => 2)-
Create and return a new path object.
($x,$y) = $path->n_to_xy ($n)-
Return the X,Y coordinates of point number
$non the path. Points begin at 0 and if$n < 0then the return is an empty list.Fractional positions give an X,Y position along a straight line between the integer positions.
The optional
armsparameter can trace 1 to 6 copies of the curve, each arm successively advancing. $n = $path->xy_to_n ($x,$y)-
Return the point number for coordinates
$x,$y. If there's nothing at$x,$ythen returnundef.The curve can visits an
$x,$yup to three times. In the current code the smaller of the these N values is returned. Is that the best way? $n = $path->n_start()-
Return 0, the first N in the path.
OEIS
The terdragon is in Sloane's Online Encyclopedia of Integer Sequences as the turn at each line segment,
http://oeis.org/A080846 etc
A080846 -- turn 0=left, 1=right, by 120 degrees
A060236 -- turn 1=left, 2=right
A026225 -- N positions of left turns
A026179 -- N positions of right turns