Math::PlanePath::TerdragonCurve -- triangular dragon curve
use Math::PlanePath::TerdragonCurve; my $path = Math::PlanePath::TerdragonCurve->new; my ($x, $y) = $path->n_to_xy (123);
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 3
Points are a triangular grid using every second integer X,Y as per "Triangular Lattice" in Math::PlanePath.
The base figure is an "S" shape
2-----3 \ \ 0-----1
which then repeats in self-similar style, so N=3 to N=6 is a copy rotated +120 degrees, which is the angle of the N=1 to N=2 edge,
6 4 base figure repeats \ / \ as N=3 to N=6, \/ \ rotated +120 degrees 5 2----3 \ \ 0-----1
Then N=6 to N=9 is a plain horizontal, which is the angle of N=2 to N=3,
8-----9 base figure repeats \ as N=6 to N=9, \ no rotation 6----7,4 \ / \ \ / \ 5,2----3 \ \ 0-----1
Notice N=5 is a repeat of point X=1,Y=1 which is also N=2, and then N=7 repeats the N=4 position X=2,Y=2. Each point repeats up to 3 times. Inner points are 3 times and on the edges of the curve area up to 2 times. The first tripled point is X=1,Y=3 which can be seen above as N=8, N=11 and N=14.
The curve never crosses itself. The vertices touch as little triangular corners and no edges repeat.
The shape is the same as the GosperSide, but the turns here are by 120 degrees each whereas the GosperSide is by 60 degrees each. The extra angle here tightens up the shape.
The first step N=1 is to the right along the X axis and the path then slowly spirals anti-clockwise and progressively fatter. The end of each replication is
Nlevel = 3^level
That point is at level*30 degrees around (as reckoned with the usual Y*sqrt(3) for a triangular grid, per "Triangular Lattice" in Math::PlanePath).
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 -54,0 180
The following is points N=0 to N=3^6=729 going half-circle around to 180 degrees. The N=0 origin is marked "o" and the N=729 end marked "e".
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * e * * * * * * * * * * * * * * * * o * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
The little "S" shapes of the base figure N=0 to N=3 can be thought of as a parallelogram
2-----3 . . . . 0-----1
The "S" shapes of each 3 points make a tiling of the plane with those parallelograms
\ \ / / \ \ / / *-----*-----* *-----*-----* / / \ \ / / \ \ \ / / \ \ / / \ \ / --*-----* *-----*-----* *-----*-- / \ \ / / \ \ / / \ \ \ / / \ \ / / *-----*-----* *-----*-----* / / \ \ / / \ \ \ / / \ \ / / \ \ / --*-----* *-----o-----* *-----*-- / \ \ / / \ \ / / \ \ \ / / \ \ / / *-----*-----* *-----*-----* / / \ \ / / \ \
As per for example
http://tilingsearch.org/HTML/data23/C07A.html
The curve fills a sixth of the plane and six copies mesh together perfectly rotated by 60, 120, 180, 240 and 300 degrees. The arms parameter can choose 1 to 6 such curve arms successively advancing.
arms
For example arms => 6 begins as follows. N=0,6,12,18,etc is the first arm (the same shape as the plain curve above), then N=1,7,13,19 the second, N=2,8,14,20 the third, etc.
arms => 6
\ / \ / \ / \ / --- 8/13/31 ---------------- 7/12/30 --- / \ / \ \ / \ / \ / \ / \ / \ / --- 9/14/32 ------------- 0/1/2/3/4/5 -------------- 6/17/35 --- / \ / \ / \ / \ / \ / \ \ / \ / --- 10/15/33 ---------------- 11/16/34 --- / \ / \ / \ / \
With six arms every X,Y point is visited three times, except the origin 0,0 where all six begin. Every edge between the points is traversed once.
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::TerdragonCurve->new ()
$path = Math::PlanePath::TerdragonCurve->new (arms => 6)
Create and return a new path object.
The optional arms parameter can make 1 to 6 copies of the curve, each arm successively advancing.
($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
$n < 0
Fractional positions give an X,Y position along a straight line between the integer positions.
$n = $path->xy_to_n ($x,$y)
Return the point number for coordinates $x,$y. If there's nothing at $x,$y then return undef.
$x,$y
undef
The curve can visit an $x,$y up to three times. In the current code the smallest of the these N values is returned. Is that the best way?
@n_list = $path->xy_to_n_list ($x,$y)
Return a list of N point numbers for coordinates $x,$y. There can be none, one, two or three N's for a given $x,$y.
$n = $path->n_start()
Return 0, the first N in the path.
There's no reversals or reflections in the curve so n_to_xy() can take the digits of N either low to high or high to low applying what's in effect powers of the N=3 position. The current code goes low to high using i,j,k coordinates as described in "Triangular Calculations" in Math::PlanePath.
n_to_xy()
si = 1 # position of endpoint N=3^level sj = 0 # where level=number of digits processed sk = 0 i = 0 # position of N for digits so far processed j = 0 k = 0 loop base 3 digits of N low to high if digit == 0 i,j,k no change if digit == 1 (i,j,k) = (si-j, sj-k, sk+i) # rotate +120, add si,sj,sk if digit == 2 i -= sk # add (si,sj,sk) rotated +60 j += si k += sj (si,sj,sk) = (si - sk, # add rotated +60 sj + si, sk + sj)
The digit handling is a combination of rotate and offset,
digit==1 digit 2 rotate and offset offset at si,sj,sk rotated ^ 2------> \ \ \ *--- --1 *-- --*
The calculation can also be thought of as using w=1/2+I*sqrt(3)/2, a complex sixth root of unity. i is the real part, j in the w direction (60 degrees), and k in the w^2 direction (120 degrees). si,sj,sk increase as if multiplied by w+1.
The current code applies TerdragonMidpoint xy_to_n() to calculate six candidate N from the six edges around a point. Those N values which convert back to the target X,Y by n_to_xy() are the results for xy_to_n_list().
xy_to_n()
xy_to_n_list()
The six edges are three going towards the point and three going away. The midpoint calculation gives N-1 for the towards and N for the away. Is there a good way to tell which edge is the smallest? Or just which 3 edges lead away? It might be directions 0,2,4 for the even arms and 1,3,5 for the odd ones, but the boundary of those areas is tricky.
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 the lowest non-zero digit gives the turn
ternary lowest non-zero Turn -------- ---- 1 left 2 right
Essentially at N=3^level or N=2*3^level the turn follows the shape at that 1 or 2 point. The first and last unit step in each level are in the same direction, so the next level shape gives the turn.
2*3^k-------3^(k+1) \ \ 0-------1*3^k
The next turn, ie. the turn at position N+1, can be calculated from the ternary digits of N similarly. The lowest non-2 digit gives the turn.
ternary lowest non-2 Turn ------- ---- 0 left 1 right
If N is all 2s then the lowest non-2 is taken to be a 0 above the high end. For example N=8 is 22 ternary so considered 022 for lowest non-2 digit=0 and turn left after the segment at N=8, ie. at N=9 turn left.
The direction at N, ie. the total cumulative turn, is given by the number of 1 digits when N is written in ternary,
direction = (count 1s in ternary N) * 120 degrees
For example N=12 is ternary 110 which has two 1s so the cumulative turn at that point is 2*120=240 degrees, ie. the segment N=16 to N=17 is at angle 240.
The terdragon is in Sloane's Online Encyclopedia of Integer Sequences as,
http://oeis.org/A080846 etc A060236 turn 1=left,2=right, by 120 degrees (lowest non-zero ternary digit) A137893 turn 1=left,0=right (morphism) A189640 turn 1=left,0=right (morphism, extra initial 0) A189673 turn 0=left,1=right (morphism, extra initial 0) A080846 next turn 0=left,1=right, by 120 degrees (n=0 first turn is for N=1) A038502 strip trailing ternary 0s, taken mod 3 is turn 1=left,2=right A026225 N positions of left turns, being (3*i+1)*3^j so lowest non-zero digit is a 1 A026179 N positions of right turns (except initial 1) A060032 bignum turns 1=left,2=right to 3^level A062756 total turn, count ternary 1s A005823 N positions where total turn == 0, ternary no 1s
A189673 and A026179 start with extra initial values arising from their morphism definition. That can be skipped to consider the turns starting with a left turn at N=1.
Math::PlanePath, Math::PlanePath::TerdragonRounded, Math::PlanePath::TerdragonMidpoint, Math::PlanePath::GosperSide
Math::PlanePath::DragonCurve, Math::PlanePath::R5DragonCurve
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012 Kevin Ryde
This file is part of Math-PlanePath.
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