NAME
Math::PlanePath::AlternateTerdragon -- alternate terdragon curve
SYNOPSIS
use Math::PlanePath::AlternateTerdragon;
my $path = Math::PlanePath::AlternateTerdragon->new;
my ($x, $y) = $path->n_to_xy (123);DESCRIPTION
This is the alternate terdragon curve by Davis and Knuth,
Chandler Davis and Donald Knuth, "Number Representations and Dragon Curves -- I", Journal Recreational Mathematics, volume 3, number 2 (April 1970), pages 66-81 and "Number Representations and Dragon Curves -- II", volume 3, number 3 (July 1970), pages 133-149.
Reprinted with addendum in Knuth "Selected Papers on Fun and Games", 2010, pages 571--614. http://www-cs-faculty.stanford.edu/~uno/fg.html
Points are a triangular grid using every second integer X,Y as per "Triangular Lattice" in Math::PlanePath, beginning
\ / \ /
Y=2 14,17 --- 15,24,33 --
\ / \
\ / \ /
Y=1 2 ------- 3,12 ---- 10,13,34 -- 32,35,38
\ / \ / \ / \
\ / \ / \ /
Y=0 0 -------- 1,4 ----- 5,8,11 ----- 9,36 ----
/ \
/ \
Y=-1 6 --------- 7
^ ^ ^ ^ ^ ^ ^ ^
X=0 1 2 3 4 5 6 7A segment 0 to 1 is unfolded to
2-----3
\
\
0-----1Then 0 to 3 is unfolded likewise, but the folds are the opposite way. Where 1-2 went on the left, for 3-6 goes to the right.
2-----3 2-----3
\ / \ /
\ / \ /
0----1,4----5 0----1,4---5,8----9
/ / \
/ / \
6 6-----7Successive unfolds go alternate ways. Taking two unfold at a time is segment replacement by the 0 to 9 figure (rotated as necessary). The curve never crosses itself. Vertices touch at triangular corners. Points can be visited 1, 2 or 3 times.
The two triangles have segment 4-5 between. In general points to a level N=3^k have a single segment between two blobs, for example N=0 to N=3^6=729 below. But as the curve continues it comes back to put further segments there (and a single segment between bigger blobs).
* *
* * * *
* * * *
* * * * * * *
* * * * * * * * * *
* * * * * * * * * *
* * * * * * * * * *
* * * * * * * * * * *
* * * * * * * * * *
* * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * * * * * *
O * * * * * * * * * * * * * * * * * * * * * * * * * * E
* * * * * * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * *
* * * * * * * * * *
* * * * * * * * * * *
* * * * * * * * * *
* * * * * * * * * *
* * * * * * * * * *
* * * * * * *
* * * *
* * * *
* *The top boundary extent is at an angle +60 degrees and the bottom at -30 degrees,
/ 60 deg
/
/
O-------------------
--__
--__ 30 degAn even expansion level is within a rectangle with endpoint at X=2*3^(k/2),Y=0.
Arms
The curve fills a sixth of the plane and six copies rotated by 60, 120, 180, 240 and 300 degrees mesh together perfectly. The arms parameter can choose 1 to 6 such curve arms successively advancing.
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.
\ / \ /
\ / \ /
--- 7,8,26 ----------------- 1,12,19 ---
/ \ / \
\ / \ / \ /
\ / \ / \ /
--- 3,14,21 ------------- 0,1,2,3,4,5 -------------- 6,11,24 ---
/ \ / \ / \
/ \ / \ / \
\ / \ /
---- 9,10,28 ---------------- 5,16,23 ---
/ \ / \
/ \ / \With six arms every X,Y point is visited three times, except the origin 0,0 where all six begin. Every edge between points is traversed once.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::AlternateTerdragon->new ()$path = Math::PlanePath::AlternateTerdragon->new (arms => 6)-
Create and return a new path object.
The optional
armsparameter 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
$non the path. Points begin at 0 and if$n < 0then the return is an empty list.Fractional positions give an X,Y position along a straight line between the integer positions.
$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 visit an
$x,$yup to three times.xy_to_n()returns the smallest of the these N values. @n_list = $path->xy_to_n_list ($x,$y)-
Return a list of N point numbers for coordinates
$x,$y.The origin 0,0 has
arms_count()many N since it's the starting point for each arm. Other points have up to 3 Ns for a given$x,$y. If arms=6 then every even$x,$yexcept the origin has exactly 3 Ns.
Descriptive Methods
$n = $path->n_start()-
Return 0, the first N in the path.
$dx = $path->dx_minimum()$dx = $path->dx_maximum()$dy = $path->dy_minimum()$dy = $path->dy_maximum()-
The dX,dY values on the first arm take three possible combinations, being 120 degree angles.
dX,dY for arms=1 ----- 2, 0 dX minimum = -1, maximum = +2 -1, 1 dY minimum = -1, maximum = +1 1,-1For 2 or more arms the second arm is rotated by 60 degrees so giving the following additional combinations, for a total six. This changes the dX minimum.
dX,dY for arms=2 or more ----- -2, 0 dX minimum = -2, maximum = +2 1, 1 dY minimum = -1, maximum = +1 -1,-1 $sum = $path->sumxy_minimum()$sum = $path->sumxy_maximum()-
Return the minimum or maximum values taken by coordinate sum X+Y reached by integer N values in the path. If there's no minimum or maximum then return
undef.S=X+Y is an anti-diagonal. The first arm is entirely above a line 135deg -- -45deg, per the +60deg to -30deg extents shown above. Likewise the second arm which is to 60+60=120deg. They have
sumxy_minimum = 0. More arms and allsumxy_maximumare unbounded soundef. $diffxy = $path->diffxy_minimum()$diffxy = $path->diffxy_maximum()-
Return the minimum or maximum values taken by coordinate difference X-Y reached by integer N values in the path. If there's no minimum or maximum then return
undef.D=X-Y is a leading diagonal. The first arm is entirely right of a line 45deg -- -135deg, per the +60deg to -30deg extents shown above, so it has
diffxy_minimum = 0. More arms and alldiffxy_maximumare unbounded soundef.
Level Methods
($n_lo, $n_hi) = $path->level_to_n_range($level)-
Return
(0, 3**$level), or for multiple arms return(0, $arms * 3**$level + ($arms-1)).There are 3^level segments in a curve level, so 3^level+1 points numbered from 0. For multiple arms there are arms*(3^level+1) points, numbered from 0 so n_hi = arms*(3^level+1)-1.
FORMULAS
Turn
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 at its position gives the turn. Positions are counted from 0 for the least significant digit and up from there.
turn ternary lowest non-zero digit
----- ---------------------------------------
left 1 at even position or 2 at odd position
right 2 at even position or 1 at odd positionThe flip of turn at odd positions is the "alternating" in the curve.
next turn ternary lowest non-2 digit
--------- ---------------------------------------
left 0 at even position or 1 at odd position
right 1 at even position or 0 at odd positionTotal Turn
The direction at N, ie. the total cumulative turn, is given by the 1 digits of N written in ternary.
direction = 120deg * sum / +1 if digit=1 at even position
\ -1 if digit=1 at odd positionThis is used mod 3 for n_to_dxdy().
X,Y to N
The current code is roughly the same as TerdragonCurve xy_to_n(), but with a conjugate (negate Y, reverse direction d) after each digit low to high.
X,Y Visited
When arms=6 all "even" points of the plane are visited. As per the triangular representation of X,Y this means
X+Y mod 2 == 0 "even" pointsOEIS
Sequences in Sloane's Online Encyclopedia of Integer Sequences related to the alternate terdragon include,
http://oeis.org/A156595 (etc)
A156595 next turn 0=left, 1=right (morphism)
A189715 N positions of left turns
A189716 N positions of right turns
A189717 count right turns so farSEE ALSO
Math::PlanePath, Math::PlanePath::TerdragonCurve
Math::PlanePath::DragonCurve, Math::PlanePath::AlternatePaper
HOME PAGE
http://user42.tuxfamily.org/math-planepath/index.html
LICENSE
Copyright 2018 Kevin Ryde
This file is part of Math-PlanePath.
Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.
Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
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