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 = $path->n_start()"
- Return 0, the first N in the path.
Level Methods
- "($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 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.
LICENSE
Copyright 2012, 2013, 2014, 2015, 2016 Kevin RydeThis 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/>.