trusty (3) Math::PlanePath::AlternatePaper.3pm.gz
NAME
Math::PlanePath::AlternatePaper -- alternate paper folding curve
SYNOPSIS
use Math::PlanePath::AlternatePaper;
my $path = Math::PlanePath::AlternatePaper->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This is an integer version of the alternate paper folding curve (a variation on the DragonCurve paper
folding).
8 | 128
| |
7 | 42---43/127
| | |
6 | 40---41/45--44/124
| | | |
5 | 34---35/39--38/46--47/123
| | | | |
4 | 32---33/53--36/52--37/49--48/112
| | | | | |
3 | 10---11/31--30/54--51/55--50/58--59/111
| | | | | | |
2 | 8----9/13--12/28--29/25--24/56--57/61--60/108
| | | | | | | |
1 | 2----3/7---6/14--15/27--26/18--19/23---22/62--63/107
| | | | | | | | |
Y=0 | 0-----1 4-----5 16-----17 20-----21 64---..
|
+------------------------------------------------------------
X=0 1 2 3 4 5 6 7 8
The curve visits the X axis points and the X=Y diagonal points once each and visits "inside" points
between there twice each. The first doubled point is X=2,Y=1 which is N=3 and also N=7. The segments
N=2,3,4 and N=6,7,8 have touched, but the curve doesn't cross over itself. The doubled 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 fills the eighth of the plane up to the
X=Y diagonal inclusive.
The X axis N=0,1,4,5,16,17,etc is the integers which have only digits 0,1 in base 4, or equivalently
those which have a 0 bit at each odd numbered bit position.
The X=Y diagonal N=0,2,8,10,32,etc is the integers which have only digits 0,2 in base 4, or equivalently
those which have a 0 bit at each even numbered bit position.
The X axis values are the same as on the ZOrderCurve X axis, and the X=Y diagonal is the same as the
ZOrderCurve Y axis, but in between the two are different. (See Math::PlanePath::ZOrderCurve.)
Paper Folding
The curve arises from thinking of a strip of paper folded in half alternately one way and the other, and
then unfolded so each crease is a 90 degree angle. The effect is that the curve repeats in successive
doublings turned 90 degrees and reversed.
The first segment N=0 to N=1 unfolds clockwise, pivoting at the endpoint "1",
2
-> |
unfold / |
===> | |
|
0------1 0-------1
Then that "L" shape unfolds again, pivoting at the end "2", but anti-clockwise, on the opposite side to
the first unfold,
2-------3
2 | |
| unfold | ^ |
| ===> | _/ |
| | |
0------1 0-------1 4
In general after each unfold the shape is a triangle as follows. "N" marks the N=2^k endpoint in the
shape, either bottom right or top centre.
after even number after odd number
of unfolds, of unfolds,
N=0 to N=2^even N=0 to N=2^odd
. N
/| / \
/ | / \
/ | / \
/ | / \
/ | / \
/_____N /___________\
0,0 0,0
For an even number of unfolds the triangle consists of 4 sub-parts numbered by the high digit of N in
base 4. Those sub-parts are self-similar in the direction ">", "^" etc as follows, and with a reversal
for parts 1 and 3.
+
/|
/ |
/ |
/ 2>|
+----+
/|\ 3|
/ | \ v|
/ |^ \ |
/ 0>| 1 \|
+----+----+
Arms
The "arms" parameter can choose 1 to 8 curve arms successively advancing. Each fills an eighth of the
plane. The second arm is mirrored across the X=Y leading diagonal, so
arms => 2
| | | | | |
4 | 33---31/55---25/57---23/63---64/65--
| | | | |
3 | 11---13/29---19/27---20/21---22/62--
| | | | | |
2 | 9----7/15---16/17---18/26---24/56--
| | | | |
1 | 3----4/5-----6/14---12/28---30/54--
| | | | | |
Y=0 | 0/1----2 8------10 32---
|
+------------- -------------------------
X=0 1 2 3 4
Here the even N=0,2,4,6,etc is the plain curve below the X=Y diagonals and odd N=1,3,5,7,9,etc is the
mirrored copy.
Arms 3 and 4 are the same but rotated +90 degrees and starting from X=0,Y=1. That start point ensures
each edge between integer points is traversed just once.
arms => 4
| | | | |
--34/35---14/30---18/21--25/57----37/53-- 3
| | | | |
--15/31---10/11----6/17--13/29----32/33-- 2
| | | | |
--19 7-----2/3/5---8/9-----12/28-- 1
| | |
0/1-----4 16-- <- Y=0
-----------------------------------------
-1 -2 X=0 1 2
Points N=0,4,8,12,etc is the plain curve, N=1,5,9,13,etc the second mirrored arm, N=2,6,10,14,etc is arm
3 which is the plain curve rotated +90, and N=3,7,11,15,etc the rotated and mirrored.
Arms 5 and 6 start at X=-1,Y=1, and arms 7 and 8 start at X=-1,Y=0 so they too traverse each edge once.
With a full 8 arms each point is visited twice except for the four start points which are three times.
arms => 8
| | | | | |
--75/107--66/67---26/58---34/41---49/113--73/105-- 3
| | | | | |
--51/115---27/59---18/19--10/33---25/57---64/65-- 2
| | | | | |
--36/43---12/35---4/5/11---2/3/9--16/17---24/56-- 1
| | | | | |
--28/60---20/21---6/7/13--0/1/15---8/39---32/47-- <- Y=0
| | | | | |
--68/69---29/61----14/37---22/23--31/63---55/119-- -1
| | | | | |
--77/109--53/117---38/45---30/62--70/71---79/111-- -2
| | | | | |
^
-2 -1 -2 X=0 1 2
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$path = Math::PlanePath::AlternatePaper->new ()"
"$path = Math::PlanePath::AlternatePaper->new (arms => $integer)"
Create and return a new path object.
"($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.
Fractional positions give an X,Y position along a straight line between the integer points.
"@n_list = $path->xy_to_n_list ($x,$y)"
Return a list of N point numbers for coordinates "$x,$y". There may be none, one or two N's for a
given "$x,$y", and for arms>=2 there are three N's at the starting X,Y points.
"$n = $path->n_start()"
Return 0, the first N in the path.
FORMULAS
Turn
At each point N the curve always turns either left or right, it never goes straight ahead. The turn is
given by the bit above the lowest 1 bit in N and whether that position is odd or even.
N = 0b...z100..00 (including possibly no trailing 0s)
^
pos, counting from 0 for least significant bit
(z bit) XOR (pos&1) Turn
------------------- ----
0 right
1 left
For example N=10 binary 0b1010 has lowest 1 bit at 0b__1_ and the bit above that is a 0 at even number
pos=2, so turn to the right.
Next Turn
The bits also give the turn after next by looking at the bit above the lowest 0.
N = 0b...w011..11 (including possibly no trailing 1s)
^
pos, counting from 0 for least significant bit
(w bit) XOR (pos&1) Next Turn
------------------- ---------
0 right
1 left
For example at N=10 binary 0b1010 the lowest 0 is the least significant bit, and above that is a 1 at odd
pos=1, so at N=10+1=11 turn right. This works simply because w011..11 when incremented becomes w100..00
which is the "z" form above.
The inversion at odd bit positions can be applied with an xor 0b1010..1010. If that's done then the turn
calculation is the same as the DragonCurve (see "Turn" in Math::PlanePath::DragonCurve).
Total Turn
The total turn can be calculated from the segment replacements resulting from the bits of N.
each bit of N from high to low
when plain state
0 -> no change
1 -> turn left if even bit pos or turn right if odd bit pos
and go to reversed state
when reversed state
1 -> no change
0 -> turn left if even bit pos or turn right if odd bit pos
and go to plain state
(bit positions numbered from 0 for the least significant bit)
This is similar to the DragonCurve (see "Total Turn" in Math::PlanePath::DragonCurve) except the turn is
either left or right according to an odd or even bit position of the transition, instead of always left
for the DragonCurve.
dX,dY
Since there's always a turn either left or right, never straight ahead, the X coordinate changes, then
the Y, alternately.
N=0
dX 1 0 1 0 1 0 -1 0 1 0 1 0 -1 0 1 0 ...
dY 0 1 0 -1 0 1 0 1 0 1 0 -1 0 -1 0 -1 ...
X changes when N is even, Y changes when N is odd. Each change is either +1 or -1. The changes are the
Golay-Rudin-Shapiro sequence, which is a parity of the count of adjacent 11 bit pairs.
In the total turn above it can be seen that if the 0->1 transition is at an odd position and 1->0
transition at an even position then there's a turn to the left followed by a turn to the right for no net
change. Likewise an even and an odd. This means runs of 1 bits with an odd length have no effect on the
direction. Runs of even length on the other hand are a left followed by a left, or a right followed by a
right, for 180 degrees, which negates the dX change. Thus
if N even then dX = (-1)^(count even length runs of 1 bits in N)
if N odd then dX = 0
This (-1)^count is related to the Golay-Rudin-Shapiro sequence,
GRS = (-1) ^ (count of adjacent 11 bit pairs in N)
= (-1) ^ count_1_bits(N & (N>>1))
= / +1 if (N & (N>>1)) even parity
\ -1 if (N & (N>>1)) odd parity
The GRS is +1 on an odd length run of 1 bits, for example a run 111 has two 11 bit pairs. The GRS is -1
on an even length run, for example 1111 has three 11 bit pairs. So modulo 2 the power in the GRS is the
same as the count of even length runs and therefore
dX = / GRS(N) if N even
\ 0 if N odd
For dY the total turn and odd/even runs of 1s is the same 180 degree changes, except N is odd for a Y
change so the least significant bit is 1 and there's no return to "plain" state. If this lowest run of
1s starts on an even position (an odd number of 1s) then it's a turn left for +1. Conversely if the run
started at an odd position (an even number of 1s) then a turn right for -1. The result for this last run
is the same "negate if even length" as the rest of the GRS, just for a slightly different reason.
dY = / 0 if N even
\ GRS(N) if N odd
dX,dY Pair
At a consecutive pair of points N=2k and N=2k+1 the dX and dY can be expressed together in terms of
GRS(k) as
dX = GRS(2k)
= GRS(k)
dY = GRS(2k+1)
= GRS(k) * (-1)^k
= / GRS(k) if k even
\ -GRS(k) if k odd
For dY reducing 2k+1 to k drops a 1 bit from the low end. If the second lowest bit is also a 1 then they
were a "11" bit pair which is lost from GRS(k). The factor (-1)^k adjusts for that, being +1 if k even
or -1 if k odd.
dSum
From the dX and dY formulas above it can be seen that their sum is simply GRS(N),
dSum = dX + dY = GRS(N)
The sum X+Y is a numbering of anti-diagonal lines,
| \ \ \
|\ \ \ \
| \ \ \ \
|\ \ \ \ \
| \ \ \ \ \
|\ \ \ \ \ \
+------------
0 1 2 3 4 5
The curve steps each time either up to the next or back to the previous according to dSum=GRS(N).
The way the curve visits edge outer X,Y points once each and inner X,Y points twice each means an anti-
diagonal d=X+Y is visited a total of d many times. The diagonal has floor(d/2)+1 many points. When d is
odd the first is visited once and the rest visited twice. When d is even the X=Y point is only visited
once. In each case the total is d many visits.
The way the coordinate sum d=X+Y occurs d many times is a geometric interpretation to the way the
cumulative GRS sequence has each value k occurring k many times. (See
Math::NumSeq::GolayRudinShapiroCumulative.)
OEIS
The alternate paper folding curve is in Sloane's Online Encyclopedia of Integer Sequences as
<http://oeis.org/A106665> (etc)
A106665 next turn 1=left,0=right, a(0) is turn at N=1
A209615 turn 1=left,-1=right
A020985 Golay/Rudin/Shapiro sequence +1,-1
dX and dY alternately
dSum, change in X+Y
A020986 Golay/Rudin/Shapiro cumulative
X coordinate (undoubled)
X+Y coordinate sum
A020990 Golay/Rudin/Shapiro * (-1)^n cumulative
Y coordinate (undoubled)
X-Y diff, starting from N=1
A020987 GRS with values 0,1 instead of +1,-1
Since the X and Y coordinates each change alternately, each coordinate appears twice, for instance
X=0,1,1,2,2,3,3,2,2,etc. A020986 and A020990 are "undoubled" X and Y in the sense of just one copy of
each of those paired values.
A077957 Y at N=2^k, being alternately 0 and 2^(k/2)
A000695 N on X axis, base 4 digits 0,1 only
A062880 N on diagonal, base 4 digits 0,2 only
A022155 N positions of left or down segment,
being GRS < 0,
ie. dSum < 0 so move to previous anti-diagonal
A203463 N positions of up or right segment,
being GRS > 0,
ie. dSum > 0 so move to next anti-diagonal
A020991 N-1 of first time on X+Y=k anti-diagonal
A212591 N-1 of last time on X+Y=k anti-diagonal
A093573 N-1 of points on the anti-diagonals d=X+Y,
by ascending N-1 value within each diagonal
A020991 etc have values N-1, ie. the numbering differs by 1 from the N here, since they're based on the
A020986 cumulative GRS starting at n=0 for value GRS(0). This matches the turn sequence A106665 starting
at n=0 for the first turn, whereas for the path here that's N=1.
SEE ALSO
Math::PlanePath, Math::PlanePath::AlternatePaperMidpoint
Math::PlanePath::DragonCurve, Math::PlanePath::CCurve, Math::PlanePath::HIndexing,
Math::PlanePath::ZOrderCurve
Math::NumSeq::GolayRudinShapiro, Math::NumSeq::GolayRudinShapiroCumulative
Michel Mendes France and G. Tenenbaum, "Dimension des Courbes Planes, Papiers Plies et Suites de Rudin-
Shapiro", Bulletin de la S.M.F., volume 109, 1981, pages 207-215.
<http://www.numdam.org/item?id=BSMF_1981__109__207_0>
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE
Copyright 2011, 2012, 2013 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/>.