Provided by: libmath-planepath-perl_113-1_all bug

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/>.