1Math::PlanePath::KochPeUaskesr(3C)ontributed Perl DocumeMnattaht:i:oPnlanePath::KochPeaks(3)
2
3
4

NAME

6       Math::PlanePath::KochPeaks -- Koch curve peaks
7

SYNOPSIS

9        use Math::PlanePath::KochPeaks;
10        my $path = Math::PlanePath::KochPeaks->new;
11        my ($x, $y) = $path->n_to_xy (123);
12

DESCRIPTION

14       This path traces out concentric peaks made from integer versions of the
15       self-similar "KochCurve" at successively greater replication levels.
16
17                                      29                                 9
18                                     /  \
19                             27----28    30----31                        8
20                               \              /
21                    23          26          32          35               7
22                   /  \        /              \        /  \
23           21----22    24----25                33----34    36----37      6
24             \                                                  /
25              20                                              38         5
26             /                                                  \
27           19----18                                        40----39      4
28                   \                                      /
29                    17                 8                41               3
30                   /                 /  \                 \
31           15----16           6---- 7     9----10          42----43      2
32             \                 \              /                 /
33              14                 5     2    11                44         1
34             /                 /     /  \     \                 \
35           13                 4     1    3     12                45  <- Y=0
36
37                                       ^
38           -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7  8  9 ...
39
40       The initial figure is the peak N=1,2,3 then for the next level each
41       straight side expands to 3x longer with a notch in the middle like N=4
42       through N=8,
43
44                                         *
45                                        / \
46             *---*     becomes     *---*   *---*
47
48       The angle is maintained in each replacement so
49
50                                         *
51                                        /
52                                   *---*
53                                    \
54               *                     *
55              /        becomes      /
56             *                     *
57
58       For example the segment N=1 to N=2 becomes N=4 to N=8, or in the next
59       level N=5 to N=6 becomes N=17 to N=21.
60
61       The X,Y coordinates are arranged as integers on a square grid.  The
62       result is flattened triangular segments with diagonals at a 45 degree
63       angle.
64
65       Unlike other triangular grid paths "KochPeaks" uses the "odd" squares,
66       with one of X,Y odd and the other even.  This means the rotation
67       formulas etc described in "Triangular Lattice" in Math::PlanePath don't
68       apply directly.
69
70   Level Ranges
71       Counting the innermost N=1 to N=3 peak as level 0, each peak is
72
73           Nstart = level + (2*4^level + 1)/3
74           Nend   = level + (8*4^level + 1)/3
75           points = Nend-Nstart+1 = 2*4^level + 1
76
77       For example the outer peak shown above is level 2 starting at
78       Nstart=2+(2*4^2+1)/3=13 through to Nend=2+(8*4^2+1)/3=45 with
79       points=2*4^2+1=33 inclusive (45-13+1=33).  The X width at a given level
80       is the endpoints at
81
82           Xlo = -(3^level)
83           Xhi = +(3^level)
84
85       For example the level 2 above runs from Xlo=-9 to Xhi=+9.  The highest
86       Y is the centre peak half-way through the level at
87
88           Ypeak = 3^level
89           Npeak = level + (5*4^level + 1)/3
90
91       For example the level 2 outer peak above is Ypeak=3^2=9 at
92       N=2+(5*4^2+1)/3=29.  For each level the Xlo,Xhi and Ypeak extents grow
93       by a factor of 3.
94
95       The triangular notches in each segment are not big enough to go past
96       the Xlo and Xhi end points.  The new triangular part can equal the
97       ends, such as N=6 or N=19, but not go beyond.
98
99       In general a segment like N=5 to N=6 which is at the Xlo end will
100       expand to give two such segments and two points at the limit in the
101       next level, as for example N=5 to N=6 expands to N=19,20 and N=20,21.
102       So the count of points at Xlo doubles each time,
103
104           CountLo = 2^level
105           CountHi = 2^level      same at Xhi
106

FUNCTIONS

108       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
109       classes.
110
111       "$path = Math::PlanePath::KochPeaks->new ()"
112           Create and return a new path object.
113
114       "($x,$y) = $path->n_to_xy ($n)"
115           Return the X,Y coordinates of point number $n on the path.  Points
116           begin at 0 and if "$n < 0" then the return is an empty list.
117
118           Fractional $n gives an X,Y position along a straight line between
119           the integer positions.
120
121   Level Methods
122       "($n_lo, $n_hi) = $path->level_to_n_range($level)"
123           Return per "Level Ranges" above,
124
125               ((2 * 4**$level + 1)/3 + $level,
126                (8 * 4**$level + 1)/3 + $level)
127

FORMULAS

129   Rectangle to N Range
130       The baseline for a given level is along a diagonal X+Y=3^level or
131       -X+Y=3^level.  The containing level can thus be found as
132
133           level = floor(log3( Xmax + Ymax ))
134           with Xmax as maximum absolute value, max(abs(X))
135
136       The endpoint in a level is simply 1 before the start of the next, so
137
138            Nlast = Nstart(level+1) - 1
139                  = (level+1) + (2*4^(level+1) + 1)/3 - 1
140                  = level + (8*4^level + 1)/3
141
142       Using this Nlast is an over-estimate of the N range needed, but an easy
143       calculation.
144
145       It's not too difficult to work down for an exact range, by considering
146       which parts of the curve might intersect a rectangle.  But some
147       backtracking and level descending is necessary because a rectangle
148       might extend into the empty part of a notch and so be past its baseline
149       but not intersect any.  There's plenty of room for a rectangle to
150       intersect nothing at all too.
151

SEE ALSO

153       Math::PlanePath, Math::PlanePath::KochCurve,
154       Math::PlanePath::KochSnowflakes, Math::PlanePath::PeanoCurve,
155       Math::PlanePath::HilbertCurve
156

HOME PAGE

158       <http://user42.tuxfamily.org/math-planepath/index.html>
159

LICENSE

161       Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020
162       Kevin Ryde
163
164       Math-PlanePath is free software; you can redistribute it and/or modify
165       it under the terms of the GNU General Public License as published by
166       the Free Software Foundation; either version 3, or (at your option) any
167       later version.
168
169       Math-PlanePath is distributed in the hope that it will be useful, but
170       WITHOUT ANY WARRANTY; without even the implied warranty of
171       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
172       General Public License for more details.
173
174       You should have received a copy of the GNU General Public License along
175       with Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.
176
177
178
179perl v5.34.0                      2021-07-22     Math::PlanePath::KochPeaks(3)
Impressum