SYNOPSIS
use Math::PlanePath::R5DragonMidpoint;
my $path = Math::PlanePath::R5DragonMidpoint>new;
my ($x, $y) = $path>n_to_xy (123);
DESCRIPTION
This is midpoints of the R5 dragon curve by Jorg Arndt,
3130 11   32 29 10   5150 353433 282726 9     52 49 363738 232425 8     555453 484746 414039 22 7     565758 6364 45 42 192021 6       8180 59 62 65 4443 181716 1110 5         82 79 6061 666768 15 12 9 4       ..83 787776 717069 1413 8 7 6 3    75 72 3 4 5 2    7473 2 1  0 1 < Y=0 ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ 10 9 8 7 6 5 4 3 2 1 X=0 1 2 3
The points are the middle of each edge of the "R5DragonCurve", rotated 45 degrees, shrunk by sqrt(2). and shifted to the origin.
*11* *7* R5DragonCurve     and its midpoints 12 10 8 6     *17*13*9*5*     18 16 14 4     ..* *15* *3*  2  +1*
Arms
Multiple copies of the curve can be selected, each advancing successively. Like the main "R5DragonCurve" this midpoint curve covers 1/4 of the plane and 4 arms rotated by 0, 90, 180, 270 degrees mesh together perfectly. With 4 arms all integer X,Y points are visited."arms => 4" begins as follows. N=0,4,8,12,16,etc is the first arm (the same shape as the plain curve above), then N=1,5,9,13,17 the second, N=2,6,10,14 the third, etc.
arms=>4 7680... 6  726864 4440 5    2521 60 48 36 4      29 17 5652 322824 7579 3      413733 13 9 5 121620 71 83 2      454953 6 2 1 8 596367 ... 1     ... 656157 10 3 0 4 555147 < Y=0      81 69 221814 71115 353943 1      7773 263034 5458 19 31 2      38 50 62 2327 3    4246 667074 4  ...8278 5 ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ 6 5 4 3 2 1 X=0 1 2 3 4 5
FUNCTIONS
See ``FUNCTIONS'' in Math::PlanePath for behaviour common to all path classes. "$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 = $path>n_start()"
 Return 0, the first N in the path.
Level Methods
 "($n_lo, $n_hi) = $path>level_to_n_range($level)"

Return "(0, 5**$level  1)", or for multiple arms return "(0, $arms *
5**$level  1)".
There are 5^level segments comprising the curve, or arms*5^level when multiple arms, numbered starting from 0.
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 base5 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) * (12i) / 5 = ((Xm+2*Ym)/5, (Ym2*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 4digit, 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.
LICENSE
Copyright 2012, 2013, 2014, 2015, 2016 Kevin RydeThis file is part of MathPlanePath.
MathPlanePath 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.
MathPlanePath 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 MathPlanePath. If not, see <http://www.gnu.org/licenses/>.