Provided by: libmath-planepath-perl_129-1_all 
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 7
A segment 0 to 1 is unfolded to
2-----3
\
\
0-----1
Then 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-----7
Successive 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 deg
An 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 ()"
"$path = Math::PlanePath::AlternateTerdragon->new (arms => 6)"
Create and return a new path object.
The optional "arms" parameter can make 1 to 6 copies of the curve, each arm
successively advancing.
"($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.
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,$y" then
return "undef".
The curve can visit an "$x,$y" up 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,$y"
except the origin has exactly 3 Ns.
Descriptive Methods
"$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 = $path->dy_maximum()"
The dX,dY values on the first arm take three possible combinations, being 120 degree
angles.
dX,dY for arms=1
-----
2, 0 dX minimum = -1, maximum = +2
-1, 1 dY minimum = -1, maximum = +1
1,-1
For 2 or more arms the second arm is rotated by 60 degrees so giving the following
additional combinations, for a total six. This changes the dX minimum.
dX,dY for arms=2 or more
-----
-2, 0 dX minimum = -2, maximum = +2
1, 1 dY minimum = -1, maximum = +1
-1,-1
"$sum = $path->sumxy_minimum()"
"$sum = $path->sumxy_maximum()"
Return the minimum or maximum values taken by coordinate sum X+Y reached by integer N
values in the path. If there's no minimum or maximum then return "undef".
S=X+Y is an anti-diagonal. The first arm is entirely above a line 135deg -- -45deg,
per the +60deg to -30deg extents shown above. Likewise the second arm which is to
60+60=120deg. They have "sumxy_minimum = 0". More arms and all "sumxy_maximum" are
unbounded so "undef".
"$diffxy = $path->diffxy_minimum()"
"$diffxy = $path->diffxy_maximum()"
Return the minimum or maximum values taken by coordinate difference X-Y reached by
integer N values in the path. If there's no minimum or maximum then return "undef".
D=X-Y is a leading diagonal. The first arm is entirely right of a line 45deg --
-135deg, per the +60deg to -30deg extents shown above, so it has "diffxy_minimum = 0".
More arms and all "diffxy_maximum" are unbounded so "undef".
Level Methods
"($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 position
The 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 position
Total 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 position
This 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" points
OEIS
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 far
HOUSE OF GRAPHS
House of Graphs entries for the alternate terdragon curve as a graph include
<https://hog.grinvin.org/ViewGraphInfo.action?id=19655> etc
19655 level=0 (1-segment path)
594 level=1 (3-segment path)
30397 level=2
30399 level=3
33575 level=4
33577 level=5
SEE 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, 2019, 2020 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.
You should have received a copy of the GNU General Public License along with Math-
PlanePath. If not, see <http://www.gnu.org/licenses/>.