Kevin Ryde > Math-PlanePath-116 > Math::PlanePath::R5DragonMidpoint

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Module Version: 116   Source   Latest Release: Math-PlanePath-117

NAME ^

Math::PlanePath::R5DragonMidpoint -- R5 dragon curve midpoints

SYNOPSIS ^

 use Math::PlanePath::R5DragonMidpoint;
 my $path = Math::PlanePath::R5DragonMidpoint->new;
 my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION ^

This is midpoints of the R5 dragon curve by Jorg Arndt,

                                       31--30                       11
                                        |   |
                                       32  29                       10
                                        |   |
               51--50          35--34--33  28--27--26                9
                |   |           |                   |
               52  49          36--37--38  23--24--25                8
                |   |                   |   |
       55--54--53  48--47--46  41--40--39  22                        7
        |                   |   |           |
       56--57--58  63--64  45  42  19--20--21                        6
                |   |   |   |   |   |
       81--80  59  62  65  44--43  18--17--16  11--10                5
        |   |   |   |   |                   |   |   |
       82  79  60--61  66--67--68          15  12   9                4
        |   |                   |           |   |   |
    ..-83  78--77--76  71--70--69          14--13   8-- 7-- 6        3
                    |   |                                   |
                   75  72                           3-- 4-- 5        2
                    |   |                           |
                   74--73                           2                1
                                                    |
                                                0-- 1           <- Y=0

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

The points are the middle of each edge of the R5DragonCurve, rotated -45 degrees, shrunk by sqrt(2). and shifted to the origin.

              *--11--*     *--7--*     R5DragonCurve
              |      |     |     |     and its midpoints
             12     10     8     6
              |      |     |     |
       *--17--*--13--*--9--*--5--*
       |      |      |     |
      18     16     14     4
       |      |      |     |
    ..-*      *--15--*     *--3--*
                                 |
                                 2
                                 |
                           +--1--*

Arms

Multiple copies of the curve can be selected, each advancing successively. Like the main R5DragonCurve this midpoint curve covers 1/4 of the plane and 4 arms rotated by 0, 90, 180, 270 degrees mesh together perfectly. With 4 arms all integer X,Y points are visited.

arms => 4 begins as follows. N=0,4,8,12,16,etc is the first arm (the same shape as the plain curve above), then N=1,5,9,13,17 the second, N=2,6,10,14 the third, etc.

    arms=>4     76--80-...                                6
                 |
                72--68--64  44--40                        5
                         |   |   |
                25--21  60  48  36                        4
                 |   |   |   |   |
                29  17  56--52  32--28--24  75--79        3
                 |   |                   |   |   |
        41--37--33  13-- 9-- 5  12--16--20  71  83        2
         |                   |   |           |   |
        45--49--53   6-- 2   1   8  59--63--67  ...       1
                 |   |           |   |
    ... 65--61--57  10   3   0-- 4  55--51--47        <- Y=0
     |   |           |   |                   |
    81  69  22--18--14   7--11--15  35--39--43           -1
     |   |   |                   |   |
    77--73  26--30--34  54--58  19  31                   -2
                     |   |   |   |   |
                    38  50  62  23--27                   -3
                     |   |   |
                    42--46  66--70--74                   -4
                                     |
                            ...-82--78                   -5

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

FUNCTIONS ^

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

$path = Math::PlanePath::R5DragonMidpoint->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.

FORMULAS ^

X,Y to N

An X,Y point can be turned into N by dividing out digits of a complex base 1+2i. At each step the low base-5 digit is formed from X,Y and an adjustment applied to move X,Y to a multiple of 1+2i ready to divide out.

A 10x10 table is used for the digit and adjustments, indexed by Xmod10 and Ymod10. There's probably an a*X+b*Y mod 5 or mod 20 for a smaller table. But in any case once the adjustment is found the result is

    Ndigit = digit_table[X mod 10, Y mod 10]  # low to high
    Xm = X + Xadj_table [X mod 10, Y mod 10]
    Ym = Y + Yadj_table [X mod 10, Y mod 10]

    new X,Y = (Xm,Ym) / (1+2i)
            = (Xm,Ym) * (1-2i) / 5
            = ((Xm+2*Ym)/5, (Ym-2*Xm)/5)

These X,Y reductions eventually reach one of the starting 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 digits must be flipped 4-digit, so 0,1,2,3,4 -> 4,3,2,1,0. The arm number and hence whether this flip is needed is not known until reaching the endpoint.

    if arm odd
    then  N = 5^numdigits - 1 - N

If only some of the arms are of interest then reaching one of the other arm numbers means the original X,Y was outside the desired curve.

SEE ALSO ^

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

Math::PlanePath::DragonMidpoint, Math::PlanePath::TerdragonMidpoint

HOME PAGE ^

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

LICENSE ^

Copyright 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/>.

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