Provided by: libmath-planepath-perl_125-1_all bug

NAME

       Math::PlanePath::DragonMidpoint -- dragon curve midpoints

SYNOPSIS

        use Math::PlanePath::DragonMidpoint;
        my $path = Math::PlanePath::DragonMidpoint->new;
        my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

       This is the midpoint of each segment of the dragon curve of Heighway, Harter, et al, per
       Math::PlanePath::DragonCurve.

                           17--16           9---8                5
                            |   |           |   |
                           18  15          10   7                4
                            |   |           |   |
                           19  14--13--12--11   6---5---4        3
                            |                           |
                           20--21--22                   3        2
                                    |                   |
           33--32          25--24--23                   2        1
            |   |           |                           |
           34  31          26                       0---1    <- Y=0
            |   |           |
           35  30--29--28--27                                   -1
            |
           36--37--38  43--44--45--46                           -2
                    |   |           |
                   39  42  49--48--47                           -3
                    |   |   |
                   40--41  50                                   -4
                            |
                           51                                   -5
                            |
                           52--53--54                           -6
                                    |
           ..--64          57--56--55                           -7
                |           |
               63          58                                   -8
                |           |
               62--61--60--59                                   -9

            ^   ^   ^   ^   ^   ^   ^   ^   ^   ^   ^   ^
           -10 -9  -8  -7  -6  -5  -4  -3  -2  -1  X=0  1

       The dragon curve begins as follows.  The midpoints of each segment are numbered starting
       from 0,

            +--8--+     +--4--+
            |     |     |     |
            9     7     5     3
            |     |     |     |                               |
            +-10--+--6--+     +--2--+       rotate 45 degrees |
                  |                 |                         v
                 11                 1
                  |                 |
            +-12--+           *--0--+       * = Origin
            |
           ...

       These midpoints are on fractions X=0.5,Y=0, X=1,Y=0.5, etc.  For this "DragonMidpoint"
       path they're turned clockwise 45 degrees and shrunk by sqrt(2) to be integer X,Y values a
       unit apart and initial direction to the right.

       The midpoints are distinct X,Y positions because the dragon curve traverses each edge only
       once.

       The dragon curve is self-similar in 2^level sections due to its unfolding.  This can be
       seen in the midpoints too as for example above N=0 to N=16 is the same shape as N=16 to
       N=32, with the latter rotated 90 degrees and in reverse.

       For reference, Knuth in "Diamonds and Dragons" has a different numbering for segment
       midpoints where the dragon orientation is unchanged and instead multiply by 2 to have
       midpoints as integers.  For example the first dragon midpoint at X=1/2,Y=0 is doubled out
       to X=1,Y=0.  That can be obtained from the path here by

           KnuthX = X - Y + 1
           KnuthY = X + Y

   Arms
       Like the "DragonCurve" the midpoints fill a quarter of the plane and four copies mesh
       together perfectly when rotated by 90, 180 and 270 degrees.  The "arms" parameter can
       choose 1 to 4 curve arms, successively advancing.

       For example "arms => 4" begins as follows, with N=0,4,8,12,etc being the first arm (the
       same as the plain curve above), N=1,5,9,13 the second, N=2,6,10,14 the third and
       N=3,7,11,15 the fourth.

           arms => 4

                           ...-107-103  83--79--75--71             6
                                     |   |           |
            68--64          36--32  99  87  59--63--67             5
             |   |           |   |   |   |   |
            72  60          40  28  95--91  55                     4
             |   |           |   |           |
            76  56--52--48--44  24--20--16  51                     3
             |                           |   |
            80--84--88  17--13---9---5  12  47--43--39 ...         2
                     |   |           |   |           |  |
           100--96--92  21   6---2   1   8  27--31--35 106         1
             |           |   |           |   |          |
           104  33--29--25  10   3   0---4  23  94--98-102    <- Y=0
             |   |           |   |           |   |
           ...  37--41--45  14   7--11--15--19  90--86--82        -1
                         |   |                           |
                        49  18--22--26  46--50--54--58  78        -2
                         |           |   |           |   |
                        53  89--93  30  42          62  74        -3
                         |   |   |   |   |           |   |
                65--61--57  85  97  34--38          66--70        -4
                 |           |   |
                69--73--77--81 101-105-...                        -5

                                     ^
            -6  -5  -4  -3  -2  -1  X=0  1   2   3   4   5

       With four arms like this every X,Y point is visited exactly once, because four arms of the
       "DragonCurve" traverse every edge exactly once.

   Tiling
       Taking pairs of adjacent points N=2k and N=2k+1 gives little rectangles with the following
       tiling of the plane repeating in 4x4 blocks.

                +---+---+---+-+-+---+-+-+---+
                |   | | |   | | |   | | |   |
                +---+ | +---+ | +---+ | +---+
                |   | | |9 8| | |   | | |   |
                +-+-+---+-+-+-+-+-+-+-+-+-+-+
                | | |   | |7|   | | |   | | |
                | | +---+ | +---+ | +---+ | |
                | | |   | |6|5 4| | |   | | |
                +---+-+-+-+-+-+-+-+-+-+-+-+-+
                |   | | |   | |3|   | | |   |
                +---+ | +---+ | +---+ | +---+
                |   | | |   | |2|   | | |   |
                +-+-+-+-+-+-+-+-+-+-+-+-+-+-+
                | | |   | | |0 1| | |   | | |   <- Y=0
                | | +---+ | +---+ | +---+ | |
                | | |   | | |   | | |   | | |
                +-+-+-+-+-+-+-+-+-+-+-+-+-+-+
                |   | | |   | | |   | | |   |
                +---+ | +---+ | +---+ | +---+
                |   | | |   | | |   | | |   |
                +---+-+-+---+-+-+---+-+-+---+
                             ^
                            X=0

       The pairs follow this pattern both for the main curve N=0 etc shown, and also for the
       rotated copies per "Arms" above.  This tiling is in the tilingsearch database as

           <http://tilingsearch.org/HTML/data24/K02A.html>

       Taking pairs N=2k+1 and N=2k+2, being each odd N and its successor, gives a regular
       pattern too, but this time repeating in blocks of 16x16.

           |||--||||||--||--||--||||||--||||||--||||||--||||||--||||||--|||
           |||--||||||--||--||--||||||--||||||--||||||--||||||--||||||--|||
           -||------||------||------||------||------||------||------||-----
           -||------||------||------||------||------||------||------||-----
           |||--||||||||||||||--||||||||||||||--||||||||||||||--|||||||||||
           |||--||||||||||||||--||||||||||||||--||||||||||||||--|||||||||||
           -----||------||------||------||------||------||------||------||-
           -----||------||------||------||------||------||------||------||-
           -||--||--||--||--||--||||||--||--||--||--||--||--||--||||||--||-
           -||--||--||--||--||--||||||--||--||--||--||--||--||--||||||--||-
           -||------||------||------||------||------||------||------||-----
           -||------||------||------||------||------||------||------||-----
           |||||||||||--||||||||||||||--||||||||||||||--||||||||||||||--|||
           |||||||||||--||||||||||||||--||||||||||||||--||||||||||||||--|||
           -----||------||------||------||------||------||------||------||-
           -----||------||------||------||------||------||------||------||-
           |||--||||||--||--||--||||||--||  ||--||||||--||--||--||||||--|||
           |||--||||||--||--||--||||||--||  ||--||||||--||--||--||||||--|||
           -||------||------||------||------||------||------||------||-----
           -||------||------||------||------||------||------||------||-----
           |||--||||||||||||||--||||||||||||||--||||||||||||||--|||||||||||
           |||--||||||||||||||--||||||||||||||--||||||||||||||--|||||||||||
           -----||------||------||------||------||------||------||------||-
           -----||------||------||------||------||------||------||------||-
           -||--||||||--||--||--||--||--||--||--||||||--||--||--||--||--||-
           -||--||||||--||--||--||--||--||--||--||||||--||--||--||--||--||-
           -||------||------||------||------||------||------||------||-----
           -||------||------||------||------||------||------||------||-----
           |||||||||||--||||||||||||||--||||||||||||||--||||||||||||||--|||
           |||||||||||--||||||||||||||--||||||||||||||--||||||||||||||--|||
           -----||------||------||------||------||------||------||------||-
           -----||------||------||------||------||------||------||------||-

   Curve from Midpoints
       Since the dragon curve always turns left or right, never straight ahead or reverse, its
       segments are alternately horizontal and vertical.  Rotated -45 degrees for the midpoints
       here this means alternately "opposite diagonal" and "leading diagonal".  They fall on X,Y
       alternately even or odd.  So the original dragon curve can be recovered from the midpoints
       by choosing leading diagonal or opposite diagonal segment according to X,Y even or odd,
       which is the same as N even or odd.

           DragonMidpoint                  dragon segment
           --------------                 -----------------
           "even" N==0 mod 2              opposite diagonal
             which is X+Y==0 mod 2 too

           "odd"  N==1 mod 2              leading diagonal
             which is X+Y==1 mod 2 too

                      /
                     3         0 at X=0,Y=0 "even", opposite diagonal
                    /          1 at X=1,Y=0 "odd", leading diagonal
                    \          etc
                     2
                      \
                \     /
                 0   1
                  \ /

FUNCTIONS

       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

       "$path = Math::PlanePath::DragonMidpoint->new ()"
           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
           positions.

       "$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 - 1)", or for multiple arms return "(0, $arms * 2**$level - 1)".

           There are 2^level segments comprising the dragon, or arms*2^level when multiple arms,
           numbered starting from 0.

FORMULAS

   X,Y to N
       An X,Y point is turned into N by dividing out digits of a complex base i+1.  This base is
       per the doubling of the "DragonCurve" at each level.  In midpoint coordinates an
       adjustment subtracting 0 or 1 must be applied to move an X,Y which is either N=2k or
       N=2k+1 to the position where dividing out i+1 gives the N=k X,Y.

       The adjustment is in a repeating pattern of 4x4 blocks.  Points N=2k and N=2k+1 both move
       to the same place corresponding to N=k multiplied by i+1.  The adjustment pattern is a
       little like the pair tiling shown above, but for some pairs both the N=2k and N=2k+1
       positions must move, it's not enough just to shift the N=2k+1 to the N=2k.

                   Xadj               Yadj
           Ymod4              Ymod4
             3 | 0 1 1 0        3 | 1 1 0 0
             2 | 1 0 0 1        2 | 1 1 0 0
             1 | 1 0 0 1        1 | 0 0 1 1
             0 | 0 1 1 0        0 | 0 0 1 1
               +--------          +--------
                 0 1 2 3            0 1 2 3
                  Xmod4              Xmod4

       The same tables work for both the main curve and for the rotated copies per "Arms" above.

           until -1<=X<=0 and 0<=Y<=1

             Xm = X - Xadj(X mod 4, Y mod 4)
             Ym = Y - Yadj(X mod 4, Y mod 4)

             new X,Y = (Xm+i*Ym) / (i+1)
                     = (Xm+i*Ym) * (1-i)/2
                     = (Xm+Ym)/2, (Ym-Xm)/2     # Xm+Ym and Ym-Xm are both even

             Nbit = Xadj xor Yadj               # bits of N low to high

       The X,Y reduction stops at one of the start points for the four arms

           X,Y endpoint   Arm        +---+---+
           ------------   ---        | 2 | 1 |  Y=1
               0, 0        0         +---+---+
               0, 1        1         | 3 | 0 |  Y=0
              -1, 1        2         +---+---+
              -1, 0        3         X=-1 X=0

       For arms 1 and 3 the N bits must be flipped 0<->1.  The arm number and hence whether this
       flip is needed is not known until reaching the endpoint.

       For bignum calculations there's no need to apply the "/2" shift in newX=(Xm+Ym)/2 and
       newY=(Ym-Xm)/2.  Instead keep a bit position which is the logical low end and pick out two
       bits from there for the Xadj,Yadj lookup.  A whole word can be dropped when the bit
       position becomes a multiple of 32 or 64 or whatever.

   Boundary
       Taking unit squares at each point, the boundary MB[k] of the resulting shape from 0 to
       N=2^k-1 inclusive can be had from the boundary B[k] of the plain dragon curve.  Taking
       points N=0 to N=2^k-1 inclusive is the midpoints of the dragon curve line segments N=0 to
       N=2^k inclusive.

           MB[k] = B[k] + 2
                 = 4, 6, 10, 18, 30, 50, 86, 146, 246, 418, 710, 1202, ...

                                    2 + x + 2*x^2
           generating function  2 * -------------
                                    1 - x - 2*x^3

       A unit square at the midpoint is a diamond on a dragon line segment

             / \
            /   \         midpoint m
           *--m--*        diamond on dragon curve line segment
            \   /
             \ /

       A boundary segment of the dragon curve has two sides of the diamond which are boundary.
       But when the boundary makes a right hand turn two such sides touch and are therefore not
       midpoint boundary.

            /^\
           / | \        right turn
           \ | //\      two diamond sides touch
            \|//  \
             *<----*
              \   /
               \ /

       The dragon curve at N=0 points East and the last segment N=2^k-1 to N=2^k is North.  Since
       the curve never overlaps itself this means that when going around the right side of the
       curve there is 1 more left turn than right turn,

           lefts - rights = 1

       The total line segments on the right is the dragon curve R[k] and there are R[k]-1 turns,
       so the total turns lefts+rights is

           lefts + rights + 1 = R[k]

       So the lefts and rights are obtained separately

           2*lefts            = R[k]       adding the two equations
           2*rights           = R[k] - 2   subtracting the two equations

       The result is then

           MR[k] = 2*R[k] - 2*rights
                 = 2*R[k] - 2*(R[k]-2)/2
                 = R[k] + 2

       A similar calculation is made on the left side of the curve.  The net turn is the same and
       so the same lefts-rights=1 and thus from the dragon curve L[k] left boundary

           ML[k] = 2*L[k] - 2*lefts
                 = 2*L[k] - 2*(L[k]/2)
                 = L[k]

       The total is then

           MB[k] = MR[k] + ML[k]
                 = R[k]+2 + L[k]
                 = B[k] + 2                 since B[k]=R[k]+L[k]

       The generating function can be had from the partial fractions form of the dragon curve
       boundary.  B[k]+2 means adding 2/(1-x) which cancels out the -2/(1-x) in gB(x).

OEIS

       The "DragonMidpoint" is in Sloane's Online Encyclopedia of Integer Sequences as

           <http://oeis.org/A073089> (etc)

           A073089   abs(dY) of n-1 to n, so 0=horizontal,1=vertical
                       (extra initial 0)
           A077860   Y at N=2^k, being Re(-(i+1)^k + i-1)
           A090678   turn=1, straight=0  (extra initial 1,1)
           A203175   boundary of unit squares N=0 to N=2^k-1, value 4 onwards

   A073089
       For A073089=abs(dY), the midpoint curve is vertical when the "DragonCurve" has a vertical
       followed by a left turn, or horizontal followed by a right turn.  "DragonCurve" verticals
       are whenever N is odd, and the turn is the bit above the lowest 0 in N (per "Turn" in
       Math::PlanePath::DragonCurve).  So

           abs(dY) = lowbit(N) XOR bit-above-lowest-zero(N)

       The n in A073089 is offset by 2 from the N numbering of the path here, so n=N+2.  The
       initial value at n=1 in A073089 has no corresponding N (it would be N=-1).

       The mod-16 definitions in A073089 express combinations of N odd/even and bit-above-low-0
       which are the vertical midpoint segments.  The recurrence a(8n+1)=a(4n+1) acts to strip
       zeros above a low 1 bit,

           n = 0b..uu0001
            -> 0b...uu001

       In terms of N=n-2 this reduces

           N = 0b..vv1111
            -> 0b...vv111

       which has the effect of seeking a lowest 0 in the range of the mod-16 conditions.

SEE ALSO

       Math::PlanePath, Math::PlanePath::DragonCurve, Math::PlanePath::DragonRounded

       Math::PlanePath::AlternatePaperMidpoint, Math::PlanePath::R5DragonMidpoint,
       Math::PlanePath::TerdragonMidpoint

HOME PAGE

       <http://user42.tuxfamily.org/math-planepath/index.html>

LICENSE

       Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017 Kevin Ryde

       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/>.