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 = $path->n_start()"
- Return 0, the first N in the path.
Level Methods¶
- "($n_lo, $n_hi) = $path->level_to_n_range($level)"
- Return "(0, 2**$level)", or for multiple arms return "(0,
$arms * 2**$level + ($arms-1))".
This is the same as "Level Methods" in
Math::PlanePath::DragonCurve. Each level is an unfold (on alternate sides
left or right).
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 Y coordinate changes, 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. Which it is follows the Golay-Rudin-Shapiro sequence which is parity odd
or even 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 outside edge X,Y points once each and inner X,Y points
twice each means an anti-diagonal s=X+Y is visited a total of s many times.
The diagonal has floor(s/2)+1 many points. When s is odd the first is visited
once and the rest visited twice. When s is even the X=Y point is only visited
once. In each case the total is s many visits.
The way the coordinate sum s=X+Y occurs s 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.)
Area¶
The area enclosed by the curve for points N=0 to N=2^k inclusive is
A[k] = (2^floor((k-1)/2) - 1) * (2^ceil((k-1)/2) - 1)
= / (2^k - 3*2^h + 2) / 2 if k odd
\ (2^k - 4*2^h + 2) / 2 if k even
where h=floor(k/2)
= 1/2*0, 0*0, 0*1, 1*1, 1*3, 3*3, 3*7, 7*7, 7*15, 15*15, ...
= 0, 0, 0, 1, 3, 9, 21, 49, 105, 225, 465, 961, ... (A027556/2)
When k is even the curve is a triangular stack with every second block along the
bottom and right sides unfilled.
*--* Y=2^h-1
| | where h=k/2
*--*--*
| |
*--*--*--*
| | | |
*--*--*--*--*
| | | |
*--*--*--*--*--*
| | | | | |
*--*--*--*--*--*--*
| | | | | |
*--*--*--*--*--*--*--*
| | | | | | | |
*--* *--* *--* *--* * Y=0
X=1 X=2^h
The area formula can be found by moving the alternating blocks in the right
column to fill the gaps in the bottom row, and moving the top half of the
triangle down to complete a rectangle
*--------*--*--*--*--*
| | | | | | height = 2^(h-1) - 1
| *--*--*--*--*--* = 2^floor((k-1)/2) - 1
| | | | | | |
| *--*--*--*--*--*--* width = 2^h - 1
| | | | | | | | = 2^ceil((k-1)/2) - 1
*--*__*--*__*--*__*--*
When k is odd the curve is a pyramid stack with every second block along the
bottom unfilled.
* Y=2^h
|
*--*--* Y=2^h-1
| | | where h=floor(k/2)
*--*--*--*--*
| | | | |
*--*--*--*--*--*--*
| | | | | | |
*--*--*--*--*--*--*--*--*
| | | | | | | | |
*--*--*--*--*--*--*--*--*--*--*
| | | | | | | | | | |
*--*--*--*--*--*--*--*--*--*--*--*--*
| | | | | | | | | | | | |
*--*--*--*--*--*--*--*--*--*--*--*--*--*--*
| | | | | | | | | | | | | | |
*--* *--* *--* *--* *--* *--* *--* *--*
X=1 X=2^h X=2^(2h)-1
This too can be rearranged, this time to make a square. The right hand half of
the bottom row fills the gaps in the left. The remaining right hand triangle
then goes above the left triangle.
* Y=2^h
|
*-----------------*--* Y=2^h - 1
| | |
| *--*--*
| | | |
| *--*--*--* height = 2^h - 1
| | | | | = 2^floor((k-1)/2)
| *--*--*--*--*
| | | | | | width = 2^h - 1
| *--*--*--*--*--* = 2^ceil((k-1)/2)
| | | | | | |
| *--*--*--*--*--*--* floor((k-1)/2) = ceil((k-1)/2)
| | | | | | | | since (k-1)/2 is an integer
*--*--*--*--*--*--*--* when k is odd
| | | | | | | |
*--*__*--*__*--*__*--*__*
X=1 X=2^h
For k=0 through k=2 there are no areas to copy this way but 2^0-1=0 in the
formula gives the desired A[0]=A[1]=A[2]=0.
Area Increment¶
The new area added between N=2^k and N=2^(k+1) is
dA[k] = A[k+1] - A[k]
= (2^floor(k/2) - 1) * 2^ceil(k/2) / 2
= (2^k - 2^ceil(k/2)) / 2
= 0, 0, 1, 2, 6, 12, 28, 56, 120, 240, 496, 992, ... (A122746)
Convex Hull Area¶
A convex hull is the smallest convex polygon which contains a given set of
points. For the alternate paper the area of the convex hull for points N=0 to
N=2^k inclusive is
HA[k] = (2^k - 1)/2
The hull is a triangle of area 2^k/2 except for an end triangle of area 1/2 at
the top for even level or right for odd level.
Right Boundary¶
The boundary length of the curve from N=0 to N=2^k on its right side is
R[k] = / 1 if k=0
| 2*2^h if k even >= 2
\ 6*2^h - 4 if k odd >= 1
where h=floor(k/2)
= 1, 2, 4, 8, 8, 20, 16, 44, 32, 92, 64, 188, 128, 380, 256, ...
For k even the right boundary is along the X axis
2^h X axis horizontals
2^h X axis indentations, if k >= 2
-----
2*2^h
For k odd the right boundary is along the X axis and then up the right side to
the top,
2*2^h - 1 X axis horizontals
2*2^h - 2 X axis indentations
2^h right slope verticals
2^h - 1 right slope horizontals
-------
6*2^h - 4
Left Boundary¶
The boundary length of the curve from N=0 to N=2^k on its left side is
L[k] = / 1 if k=0
| 4*2^h - 4 if k even >= 2
\ 2*2^h if k odd >= 1
where h=floor(k/2)
= 1, 2, 4, 4, 12, 8, 28, 16, 60, 32, 124, 64, 252, 128, 508, ...
For k even the left boundary is up the left slope then down the vertical
2^h left slope horizontals
2^h - 1 left slope verticals
2^h - 1 right edge verticals
2^h - 2 right edge indentations
-----
4*2^h - 4
For k odd the left boundary is the left slope, and this time it includes a final
vertical line segment
2^h left slope horizontals
2^h left slope verticals
-------
2*2^h
Boundary¶
The total boundary length of the curve from N=0 to N=2^k is
B[k] = L[k] + R[k] = / 6*2^h - 4 if k even
\ 8*2^h - 4 if k odd
where h=floor(k/2)
= 2, 4, 8, 12, 20, 28, 44, 60, 92, 124, 188, 252, 380, ... (2*A027383)
The special case for k=0 is eliminated since the k even 6*2^h-4 is the desired 2
when k=0, h=0.
Every enclosed unit square has all four sides traversed so by counting inside
and outside sides of the segments have 2*N = 4*A + B. This can be verified for
A[k] and B[k]
4*A[k] + B[k] = 4* / (2^h/2 - 1) * (2^h - 1) if k even
\ (2^h - 1) * (2^h - 1) if k odd
+ / 6*2^h - 4 if k even
\ 8*2^h - 4 if k odd
= / 2 * 2^h * 2^h if k even
\ 4 * 2^h * 2^h if k odd
= 2*2^k
This relation also gives a formula for B[k] using the floor and ceil pair from
A[k]
B[k] = 2*2^k - 4*A[k]
= 2*2^k - (2^floor((k+1)/2) - 2) * (2^ceil((k+1)/2) - 2)
Single Points¶
The number of single-visited points for N=0 to N=2^k inclusive is
S[k] = / 3*2^h - 1 if k even
\ 4*2^h - 1 if k odd
= 2, 3, 5, 7, 11, 15, 23, 31, 47, 63, 95, 127, ... (A052955)
The single points are all on the outer edges and those sides can be counted
easily.
The singles can also be obtained from the boundary. Each new line segment which
increases the area also increases the double points, so area=doubles. Such a
segment decreases the singles by -1 and the boundary by -2. A new line segment
which doesn't enclose new area increases the singles by +1 and the boundary by
+2. Starting from singles=1 boundary=0 means
S[N] = B[N]/2 + 1
Or with singles and doubles adding up to N+1 points the doubles=area can give
the singles from the area.
S + 2*D = N+1 N=number of segments, N+1=number of points
OEIS¶
The alternate paper folding curve is in Sloane's Online Encyclopedia of Integer
Sequences as
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.
A027556 area*2 to N=2^k
A134057 area to N=4^k
A060867 area to N=2*4^k
A122746 area increment N=2^k to N=2^(k+1)
A000225 convex hull area*2, being 2^k-1
A027383 boundary/2 to N=2^k
also boundary verticals or horizontals
(boundary is half verticals half horizontals)
A131128 boundary to N=4^k
A028399 boundary to N=2*4^k
A052955 single-visited points to N=2^k
A052940 single-visited points to N=4^k, being 3*2^n-1
arms=2
A062880 N on X axis, base 4 digits 0,2 only
arms=3
A001196 N on X axis, base 4 digits 0,3 only
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, 2014 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/>.