Provided by: libmath-planepath-perl_129-1_all bug

NAME

       Math::PlanePath::TerdragonCurve -- triangular dragon curve

SYNOPSIS

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

DESCRIPTION

       This is the 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

                     \         /       \
                  --- 26,29,32 ---------- 27                          6
                     /         \
             \      /           \
          -- 24,33,42 ---------- 22,25                                5
             /      \           /     \
                     \         /       \
                  --- 20,23,44 -------- 12,21            10           4
                     /        \        /      \        /     \
             \      /          \      /        \      /       \
               18,45 --------- 13,16,19 ------ 8,11,14 -------- 9     3
                    \          /       \      /       \
                     \        /         \    /         \
                         17              6,15 --------- 4,7           2
                                              \        /    \
                                               \      /      \
                                                 2,5 ---------- 3     1
                                                     \
                                                      \
                                           0 ----------- 1         <-Y=0

                 ^        ^        ^       ^      ^      ^      ^
                -3       -2       -1      X=0     1      2      3

       The base figure is an "S" shape

              2-----3
               \
                \
           0-----1

       which then repeats in self-similar style, so N=3 to N=6 is a copy rotated +120 degrees,
       which is the angle of the N=1 to N=2 edge,

           6      4          base figure repeats
            \   / \          as N=3 to N=6,
             \/    \         rotated +120 degrees
             5 2----3
               \
                \
           0-----1

       Then N=6 to N=9 is a plain horizontal, which is the angle of N=2 to N=3,

                 8-----9       base figure repeats
                  \            as N=6 to N=9,
                   \           no rotation
              6----7,4
               \   / \
                \ /   \
                5,2----3
                  \
                   \
              0-----1

       Notice X=1,Y=1 is visited twice as N=2 and N=5.  Similarly X=2,Y=2 as N=4 and N=7.  Each
       point can repeat up to 3 times.  "Inner" points are 3 times and on the edges up to 2
       times.  The first tripled point is X=1,Y=3 which as shown above is N=8, N=11 and N=14.

       The curve never crosses itself.  The vertices touch as triangular corners and no edges
       repeat.

       The curve turns are the same as the "GosperSide", but here the turns are by 120 degrees
       each whereas "GosperSide" is 60 degrees each.  The extra angle here tightens up the shape.

   Spiralling
       The first step N=1 is to the right along the X axis and the path then slowly spirals anti-
       clockwise and progressively fatter.  The end of each replication is

           Nlevel = 3^level

       That point is at level*30 degrees around (as reckoned with Y*sqrt(3) for a triangular
       grid).

           Nlevel      X, Y     Angle (degrees)
           ------    -------    -----
              1        1, 0        0
              3        3, 1       30
              9        3, 3       60
             27        0, 6       90
             81       -9, 9      120
            243      -27, 9      150
            729      -54, 0      180

       The following is points N=0 to N=3^6=729 going half-circle around to 180 degrees.  The N=0
       origin is marked "0" and the N=729 end is marked "E".

                                      * *               * *
                                   * * * *           * * * *
                                  * * * *           * * * *
                                   * * * * *   * *   * * * * *   * *
                                * * * * * * * * * * * * * * * * * * *
                               * * * * * * * * * * * * * * * * * * *
                                * * * * * * * * * * * * * * * * * * * *
                                   * * * * * * * * * * * * * * * * * * *
                                  * * * * * * * * * * * *   * *   * * *
                             * *   * * * * * * * * * * * *           * *
            * E           * * * * * * * * * * * * * * * *           0 *
           * *           * * * * * * * * * * * *   * *
            * * *   * *   * * * * * * * * * * * *
           * * * * * * * * * * * * * * * * * * *
            * * * * * * * * * * * * * * * * * * * *
               * * * * * * * * * * * * * * * * * * *
              * * * * * * * * * * * * * * * * * * *
               * *   * * * * *   * *   * * * * *
                        * * * *           * * * *
                       * * * *           * * * *
                        * *               * *

   Tiling
       The little "S" shapes of the base figure N=0 to N=3 can be thought of as a rhombus

              2-----3
             .     .
            .     .
           0-----1

       The "S" shapes of each 3 points make a tiling of the plane with those rhombi

               \     \ /     /   \     \ /     /
                *-----*-----*     *-----*-----*
               /     / \     \   /     / \     \
            \ /     /   \     \ /     /   \     \ /
           --*-----*     *-----*-----*     *-----*--
            / \     \   /     / \     \   /     / \
               \     \ /     /   \     \ /     /
                *-----*-----*     *-----*-----*
               /     / \     \   /     / \     \
            \ /     /   \     \ /     /   \     \ /
           --*-----*     *-----o-----*     *-----*--
            / \     \   /     / \     \   /     / \
               \     \ /     /   \     \ /     /
                *-----*-----*     *-----*-----*
               /     / \     \   /     / \     \

       Which is an ancient pattern,

           <http://tilingsearch.org/HTML/data23/C07A.html>

   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.

                         \         /             \           /
                          \       /               \         /
                       --- 8,13,31 ---------------- 7,12,30 ---
                         /        \               /         \
            \           /          \             /           \          /
             \         /            \           /             \        /
           --- 9,14,32 ------------- 0,1,2,3,4,5 -------------- 6,17,35 ---
             /         \            /           \             /        \
            /           \          /             \           /          \
                         \        /               \         /
                      --- 10,15,33 ---------------- 11,16,34 ---
                         /        \               /         \
                        /          \             /           \

       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.

       "$path = Math::PlanePath::TerdragonCurve->new ()"
       "$path = Math::PlanePath::TerdragonCurve->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

   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

       Various formulas for boundary length, area and more can be found in the author's
       mathematical write-up

           <http://user42.tuxfamily.org/terdragon/index.html>

   N to X,Y
       There's no reversals or reflections in the curve so "n_to_xy()" can take the digits of N
       either low to high or high to low and apply what is effectively powers of the N=3
       position.  The current code goes low to high using i,j,k coordinates as described in
       "Triangular Calculations" in Math::PlanePath.

           si = 1    # position of endpoint N=3^level
           sj = 0    #    where level=number of digits processed
           sk = 0

           i = 0     # position of N for digits so far processed
           j = 0
           k = 0

           loop base 3 digits of N low to high
              if digit == 0
                 i,j,k no change
              if digit == 1
                 (i,j,k) = (si-j, sj-k, sk+i)  # rotate +120, add si,sj,sk
              if digit == 2
                 i -= sk      # add (si,sj,sk) rotated +60
                 j += si
                 k += sj

              (si,sj,sk) = (si - sk,      # add rotated +60
                            sj + si,
                            sk + sj)

       The digit handling is a combination of rotate and offset,

           digit==1                   digit 2
           rotate and offset          offset at si,sj,sk rotated

                ^                          2------>
                 \
                  \                          \
           *---  --1                  *--   --*

       The calculation can also be thought of in term of w=1/2+I*sqrt(3)/2, a complex number
       sixth root of unity.  i is the real part, j in the w direction (60 degrees), and k in the
       w^2 direction (120 degrees).  si,sj,sk increase as if multiplied by w+1.

   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 gives the
       turn

          ternary lowest
          non-zero digit     turn
          --------------     -----
                1            left
                2            right

       At N=3^level or N=2*3^level the turn follows the shape at that 1 or 2 point.  The first
       and last unit step in each level are in the same direction, so the next level shape gives
       the turn.

              2*3^k-------3*3^k
                 \
                  \
           0-------1*3^k

   Next Turn
       The next turn, ie. the turn at position N+1, can be calculated from the ternary digits of
       N similarly.  The lowest non-2 digit gives the turn.

          ternary lowest
            non-2 digit       turn
          --------------      -----
                 0            left
                 1            right

       If N is all 2s then the lowest non-2 is taken to be a 0 above the high end.  For example
       N=8 is 22 ternary so considered 022 for lowest non-2 digit=0 and turn left after the
       segment at N=8, ie. at point N=9 turn left.

       This rule works for the same reason as the plain turn above.  The next turn of N is the
       plain turn of N+1 and adding +1 turns trailing 2s into trailing 0s and increments the 0 or
       1 digit above them to be 1 or 2.

   Total Turn
       The direction at N, ie. the total cumulative turn, is given by the number of 1 digits when
       N is written in ternary,

           direction = (count 1s in ternary N) * 120 degrees

       For example N=12 is ternary 110 which has two 1s so the cumulative turn at that point is
       2*120=240 degrees, ie. the segment N=16 to N=17 is at angle 240.

       The segments for digit 0 or 2 are in the "current" direction unchanged.  The segment for
       digit 1 is rotated +120 degrees.

   X,Y to N
       The current code find digits of N low to high by a remainder on X,Y to get the lowest then
       subtract and divide to unexpand.  See "unpoint" in the author's mathematical write-up for
       details.

   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

       The terdragon is in Sloane's Online Encyclopedia of Integer Sequences as,

           <http://oeis.org/A080846> (etc)

           A060236   turn 1=left,2=right, by 120 degrees
                       (lowest non-zero ternary digit)
           A137893   turn 1=left,0=right (morphism)
           A189640   turn 0=left,1=right (morphism, extra initial 0)
           A080846   next turn 0=left,1=right, by 120 degrees
                       (n=0 is turn at N=1)
           A189673   prev turn 1=left,0=right (morphism, extra initial 0)
           A038502   strip trailing ternary 0s,
                       taken mod 3 is turn 1=left,2=right
           A133162   1=segment, 2=right turn between

       A189673 and A026179 start with extra initial values arising from their morphism
       definition.  That can be skipped to consider the turns starting with a left turn at N=1.

           A026225   N positions of left turns,
                       being (3*i+1)*3^j so lowest non-zero digit is 1
           A026179   N positions of right turns (except initial 1)
                       being (3*i+2)*3^j so lowest non-zero digit is 2
           A060032   bignum turns 1=left,2=right to 3^level
           A189674   num left turns 1 to N
           A189641   num right turns 1 to N
           A189672     same

           A026141   \ dTurnLeft increment between left turns N
           A026171   /
           A026181   \ dTurnRight increment between right turns N
           A131989   /

           A062756   direction (net total turn), count ternary 1s
           A005823   N positions where direction = 0, ternary no 1s
           A023692 through A023698
                     N positions where direction = 1 to 7, ternary num 1s

           A111286   boundary length, N=0 to N=3^k, skip initial 1
           A003945   boundary/2
           A002023   boundary odd levels N=0 to N=3^(2k+1),
                     or even levels one side N=0 to N=3^(2k),
                       being 6*4^k
           A164346   boundary even levels N=0 to N=3^(2k),
                     or one side, odd levels, N=0 to N=3^(2k+1),
                       being 3*4^k
           A042950   V[k] boundary length

           A056182   area enclosed N=0 to N=3^k, being 2*(3^k-2^k)
           A081956     same
           A118004   1/2 area N=0 to N=3^(2k+1), odd levels, 9^n-4^n
           A155559   join area, being 0 then 2^k

           A099754   1/2 count distinct visited points N=0 to N=3^k

           A092236   count East segments N=0 to N=3^k-1
           A135254   count North-West segments N=0 to N=3^k-1, extra 0
           A133474   count South-West segments N=0 to N=3^k-1
           A057083   count segments diff from 3^(k-1)
           A101990   count segments same dir as middle N=0 to N=3^k-1

           A097038   num runs of 12 consecutive segments within N=0 to 3^k-1
                       each segment enclosing a new unit triangle

           A057682   level X, at N=3^level
                       also arms=2 level Y, at N=2*3^level
           A057083   level Y, at N=3^level
                       also arms=6 level X at N=6*3^level

           A057681   arms=2 level X, at N=2*3^level
                       also arms=3 level Y at 3*3^level
           A103312   same

HOUSE OF GRAPHS

       House of Graphs entries for the terdragon 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)
           21138     level=2
           21140     level=3
           33761     level=4
           33763     level=5

SEE ALSO

       Math::PlanePath, Math::PlanePath::TerdragonRounded, Math::PlanePath::TerdragonMidpoint,
       Math::PlanePath::GosperSide

       Math::PlanePath::DragonCurve, Math::PlanePath::R5DragonCurve

       Larry Riddle's Terdragon page, for boundary and area calculations of the terdragon as an
       infinite fractal <http://ecademy.agnesscott.edu/~lriddle/ifs/heighway/terdragon.htm>

HOME PAGE

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

LICENSE

       Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 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/>.