Math::PlanePath::TerdragonMidpoint -- dragon curve midpoints
use Math::PlanePath::TerdragonMidpoint; my $path = Math::PlanePath::TerdragonMidpoint->new; my ($x, $y) = $path->n_to_xy (123);
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 the TerdragonCurve segment N=3 to N=4 is X=3,Y=1 to X=2,Y=2 which is doubled out to X=6,Y=2 and X=4,Y=4, then the midpoint of those is X=5,Y=3 for N=3 here.
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 is 3 of 8 all integer X,Y points. The points used are a pattern of alternate rows with 1 of 2 points (such as the Y=7 row) and 1 of 4 points (such as the Y=8 row). Notice the pattern is the same when turned by 60 degrees or 120 degrees.
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
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.
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
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 ... / ...
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.
$n
$n < 0
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.
An X,Y point can be turned into N by dividing out digits of a complex base w+1 where
w = 1/2 + i * sqrt(3)/2 = 6th root of unity
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 of these N=0 mod 3 or N=2 mod 3 to the centre N=1 mod 3. The centre N=1 mod 3 position is always a multiple of w+1.
The angles and positions for the N triple groups 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. 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 get the table down to 6x6. But in any case once the adjustment is found the result is
Ndigit = (X + 3Y + 1) mod 3 # 0,1,2 low ternary digit 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, (3*Ym-Xm)/6)
Points not reached by the curve (ie. not the 3 of 4 triangular or 3 of 8 rectangular described above) can be detected with blank or tagged entries in the adjustment table.
The X,Y reduction stops at the midpoint of the first triple of each arm. So X=3,Y=1 which is N=1 for the first arm or 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 stop 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 so 0,1,2 becomes 2,1,0,
if arm mod 2 == 1 then N = 3**numdigits - 1 - N
Math::PlanePath, Math::PlanePath::TerdragonCurve, Math::PlanePath::TerdragonRounded
Math::PlanePath::DragonMidpoint, Math::PlanePath::R5DragonMidpoint
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012 Kevin Ryde
This file is part of Math-PlanePath.
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