1Math::PlanePath::R5DragUosneMridCpoonitnrti(b3u)ted PerlMaDtohc:u:mPelnatnaetPiaotnh::R5DragonMidpoint(3)
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6 Math::PlanePath::R5DragonMidpoint -- R5 dragon curve midpoints
7
9 use Math::PlanePath::R5DragonMidpoint;
10 my $path = Math::PlanePath::R5DragonMidpoint->new;
11 my ($x, $y) = $path->n_to_xy (123);
12
14 This is midpoints of the R5 dragon curve by Jorg Arndt,
15 31--30 11
17 | |
18 32 29 10
19 | |
20 51--50 35--34--33 28--27--26 9
21 | | | |
22 52 49 36--37--38 23--24--25 8
23 | | | |
24 55--54--53 48--47--46 41--40--39 22 7
25 | | | |
26 56--57--58 63--64 45 42 19--20--21 6
27 | | | | | |
28 81--80 59 62 65 44--43 18--17--16 11--10 5
29 | | | | | | | |
30 82 79 60--61 66--67--68 15 12 9 4
31 | | | | | |
32 ..-83 78--77--76 71--70--69 14--13 8-- 7-- 6 3
33 | | |
34 75 72 3-- 4-- 5 2
35 | | |
36 74--73 2 1
37 |
38 0-- 1 <- Y=0
39 ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
41 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3
42 The points are the middle of each edge of the "R5DragonCurve", rotated
44 -45 degrees, shrunk by sqrt(2). and shifted to the origin.
45 *--11--* *--7--* R5DragonCurve
47 | | | | and its midpoints
48 12 10 8 6
49 | | | |
50 *--17--*--13--*--9--*--5--*
51 | | | |
52 18 16 14 4
53 | | | |
54 ..-* *--15--* *--3--*
55 |
56 2
57 |
58 +--1--*
59 Arms
61 Multiple copies of the curve can be selected, each advancing
62 successively. Like the main "R5DragonCurve" this midpoint curve covers
63 1/4 of the plane and 4 arms rotated by 0, 90, 180, 270 degrees mesh
64 together perfectly. With 4 arms all integer X,Y points are visited.
65 "arms => 4" begins as follows. N=0,4,8,12,16,etc is the first arm (the
67 same shape as the plain curve above), then N=1,5,9,13,17 the second,
68 N=2,6,10,14 the third, etc.
69 arms=>4 76--80-... 6
71 |
72 72--68--64 44--40 5
73 | | |
74 25--21 60 48 36 4
75 | | | | |
76 29 17 56--52 32--28--24 75--79 3
77 | | | | |
78 41--37--33 13-- 9-- 5 12--16--20 71 83 2
79 | | | | |
80 45--49--53 6-- 2 1 8 59--63--67 ... 1
81 | | | |
82 ... 65--61--57 10 3 0-- 4 55--51--47 <- Y=0
83 | | | | |
84 81 69 22--18--14 7--11--15 35--39--43 -1
85 | | | | |
86 77--73 26--30--34 54--58 19 31 -2
87 | | | | |
88 38 50 62 23--27 -3
89 | | |
90 42--46 66--70--74 -4
91 |
92 ...-82--78 -5
93 ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
95 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5
96
98 See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
99 classes.
100 "$path = Math::PlanePath::R5DragonMidpoint->new ()"
102 Create and return a new path object.
103 "($x,$y) = $path->n_to_xy ($n)"
105 Return the X,Y coordinates of point number $n on the path. Points
106 begin at 0 and if "$n < 0" then the return is an empty list.
107 Fractional positions give an X,Y position along a straight line
109 between the integer positions.
110 "$n = $path->n_start()"
112 Return 0, the first N in the path.
113 Level Methods
115 "($n_lo, $n_hi) = $path->level_to_n_range($level)"
116 Return "(0, 5**$level - 1)", or for multiple arms return "(0, $arms
117 * 5**$level - 1)".
118 There are 5^level segments comprising the curve, or arms*5^level
120 when multiple arms, numbered starting from 0.
121
123 X,Y to N
124 An X,Y point can be turned into N by dividing out digits of a complex
125 base 1+2i. At each step the low base-5 digit is formed from X,Y and an
126 adjustment applied to move X,Y to a multiple of 1+2i ready to divide
127 out.
128 A 10x10 table is used for the digit and adjustments, indexed by Xmod10
130 and Ymod10. There's probably an a*X+b*Y mod 5 or mod 20 for a smaller
131 table. But in any case once the adjustment is found the result is
132 Ndigit = digit_table[X mod 10, Y mod 10] # low to high
134 Xm = X + Xadj_table [X mod 10, Y mod 10]
135 Ym = Y + Yadj_table [X mod 10, Y mod 10]
136 new X,Y = (Xm,Ym) / (1+2i)
138 = (Xm,Ym) * (1-2i) / 5
139 = ((Xm+2*Ym)/5, (Ym-2*Xm)/5)
140 These X,Y reductions eventually reach one of the starting points for
142 the four arms
143 X,Y endpoint Arm +---+---+
145 ------------ --- | 2 | 1 | Y=1
146 0, 0 0 +---+---+
147 0, 1 1 | 3 | 0 | Y=0
148 -1, 1 2 +---+---+
149 -1, 0 3 X=-1 X=0
150 For arms 1 and 3 the digits must be flipped 4-digit, so 0,1,2,3,4 ->
152 4,3,2,1,0. The arm number and hence whether this flip is needed is not
153 known until reaching the endpoint.
154 if arm odd
156 then N = 5^numdigits - 1 - N
157 If only some of the arms are of interest then reaching one of the other
159 arm numbers means the original X,Y was outside the desired curve.
160
162 Math::PlanePath, Math::PlanePath::R5DragonCurve
163 Math::PlanePath::DragonMidpoint, Math::PlanePath::TerdragonMidpoint
165
167 <http://user42.tuxfamily.org/math-planepath/index.html>
168
170 Copyright 2012, 2013, 2014, 2015, 2016, 2017, 2018 Kevin Ryde
171 This file is part of Math-PlanePath.
173 Math-PlanePath is free software; you can redistribute it and/or modify
175 it under the terms of the GNU General Public License as published by
176 the Free Software Foundation; either version 3, or (at your option) any
177 later version.
178 Math-PlanePath is distributed in the hope that it will be useful, but
180 WITHOUT ANY WARRANTY; without even the implied warranty of
181 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
182 General Public License for more details.
183 You should have received a copy of the GNU General Public License along
185 with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
186perl v5.32.0 2020-07-2M8ath::PlanePath::R5DragonMidpoint(3)