1Math::PlanePath::HexSpiUrsaelr(3C)ontributed Perl DocumeMnattaht:i:oPnlanePath::HexSpiral(3)
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NAME

6       Math::PlanePath::HexSpiral -- integer points around a hexagonal spiral
7

SYNOPSIS

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

DESCRIPTION

14       This path makes a hexagonal spiral, with points spread out horizontally
15       to fit on a square grid.
16
17                    28 -- 27 -- 26 -- 25                  3
18                   /                    \
19                 29    13 -- 12 -- 11    24               2
20                /     /              \     \
21              30    14     4 --- 3    10    23            1
22             /     /     /         \     \    \
23           31    15     5     1 --- 2     9    22    <- Y=0
24             \     \     \              /     /
25              32    16     6 --- 7 --- 8    21           -1
26                \     \                    /
27                 33    17 -- 18 -- 19 -- 20              -2
28                   \
29                    34 -- 35 ...                         -3
30
31            ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^
32           -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6
33
34       Each horizontal gap is 2, so for instance n=1 is at X=0,Y=0 then n=2 is
35       at X=2,Y=0.  The diagonals are just 1 across, so n=3 is at X=1,Y=1.
36       Each alternate row is offset from the one above or below.  The result
37       is a triangular lattice per "Triangular Lattice" in Math::PlanePath.
38
39       The octagonal numbers 8,21,40,65, etc 3*k^2-2*k fall on a horizontal
40       straight line at Y=-1.  In general straight lines are 3*k^2 + b*k + c.
41       A plain 3*k^2 goes diagonally up to the left, then b is a 1/6 turn
42       anti-clockwise, or clockwise if negative.  So b=1 goes horizontally to
43       the left, b=2 diagonally down to the left, b=3 diagonally down to the
44       right, etc.
45
46   Wider
47       An optional "wider" parameter makes the path wider, stretched along the
48       top and bottom horizontals.  For example
49
50           $path = Math::PlanePath::HexSpiral->new (wider => 2);
51
52       gives
53
54                                       ... 36----35                   3
55                                                   \
56                       21----20----19----18----17    34               2
57                      /                          \     \
58                    22     8---- 7---- 6---- 5    16    33            1
59                   /     /                    \     \    \
60                 23     9     1---- 2---- 3---- 4    15    32    <- Y=0
61                   \     \                          /     /
62                    24    10----11----12----13----14    31           -1
63                      \                               /
64                       25----26----27----28---29----30               -2
65
66                  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^
67                 -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7
68
69       The centre horizontal from N=1 is extended by "wider" many extra
70       places, then the path loops around that shape.  The starting point N=1
71       is shifted to the left by wider many places to keep the spiral centred
72       on the origin X=0,Y=0.  Each horizontal gap is still 2.
73
74       Each loop is still 6 longer than the previous, since the widening is
75       basically a constant amount added into each loop.
76
77   N Start
78       The default is to number points starting N=1 as shown above.  An
79       optional "n_start" can give a different start with the same shape etc.
80       For example to start at 0,
81
82           n_start => 0
83
84                    27    26    25    24                    3
85                 28    12    11    10    23                 2
86              29    13     3     2     9    22              1
87           30    14     4     0     1     8    21      <- Y=0
88              31    15     5     6     7    20   ...       -1
89                 32    16    17    18    19    38          -2
90                    33    34    35    36    37             -3
91                              ^
92           -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6
93
94       In this numbering the X axis N=0,1,8,21,etc is the octagonal numbers
95       3*X*(X+1).
96

FUNCTIONS

98       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
99       classes.
100
101       "$path = Math::PlanePath::HexSpiral->new ()"
102       "$path = Math::PlanePath::HexSpiral->new (wider => $w)"
103           Create and return a new hex spiral object.  An optional "wider"
104           parameter widens the path, it defaults to 0 which is no widening.
105
106       "($x,$y) = $path->n_to_xy ($n)"
107           Return the X,Y coordinates of point number $n on the path.
108
109           For "$n < 1" the return is an empty list, it being considered the
110           path starts at 1.
111
112       "$n = $path->xy_to_n ($x,$y)"
113           Return the point number for coordinates "$x,$y".  $x and $y are
114           each rounded to the nearest integer, which has the effect of
115           treating each $n in the path as a square of side 1.
116
117           Only every second square in the plane has an N, being those where
118           X,Y both odd or both even.  If "$x,$y" is a position without an N,
119           ie. one of X,Y odd the other even, then the return is "undef".
120

OEIS

122       Entries in Sloane's Online Encyclopedia of Integer Sequences related to
123       this path include
124
125           <http://oeis.org/A056105> (etc)
126
127           A056105    N on X axis
128           A056106    N on X=Y diagonal
129           A056107    N on North-West diagonal
130           A056108    N on negative X axis
131           A056109    N on South-West diagonal
132           A003215    N on South-East diagonal
133
134           A063178    total sum N previous row or diagonal
135           A135711    boundary length of N hexagons
136           A135708    grid sticks of N hexagons
137
138           n_start=0
139             A000567    N on X axis, octagonal numbers
140             A049451    N on X negative axis
141             A049450    N on X=Y diagonal north-east
142             A033428    N on north-west diagonal, 3*k^2
143             A045944    N on south-west diagonal, octagonal numbers second kind
144             A063436    N on WSW slope dX=-3,dY=-1
145             A028896    N on south-east diagonal
146

SEE ALSO

148       Math::PlanePath, Math::PlanePath::HexSpiralSkewed,
149       Math::PlanePath::HexArms, Math::PlanePath::TriangleSpiral,
150       Math::PlanePath::TriangularHypot
151

HOME PAGE

153       <http://user42.tuxfamily.org/math-planepath/index.html>
154

LICENSE

156       Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018 Kevin
157       Ryde
158
159       This file is part of Math-PlanePath.
160
161       Math-PlanePath is free software; you can redistribute it and/or modify
162       it under the terms of the GNU General Public License as published by
163       the Free Software Foundation; either version 3, or (at your option) any
164       later version.
165
166       Math-PlanePath is distributed in the hope that it will be useful, but
167       WITHOUT ANY WARRANTY; without even the implied warranty of
168       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
169       General Public License for more details.
170
171       You should have received a copy of the GNU General Public License along
172       with Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.
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176perl v5.32.0                      2020-07-28     Math::PlanePath::HexSpiral(3)
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