1Math::PlanePath::HeptSpUisrearlSCkoenwterdi(b3u)ted PerlMaDtohc:u:mPelnatnaetPiaotnh::HeptSpiralSkewed(3)
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NAME

6       Math::PlanePath::HeptSpiralSkewed -- integer points around a skewed
7       seven sided spiral
8

SYNOPSIS

10        use Math::PlanePath::HeptSpiralSkewed;
11        my $path = Math::PlanePath::HeptSpiralSkewed->new;
12        my ($x, $y) = $path->n_to_xy (123);
13

DESCRIPTION

15       This path makes a seven-sided spiral by cutting one corner of a square
16
17           31-30-29-28                       3
18            |         \
19           32 14-13-12 27                    2
20            |  |      \  \
21           33 15  4--3 11 26                 1
22            |  |  |   \  \  \
23           34 16  5  1--2 10 25         <- Y=0
24            |  |  |        |  |
25           35 17  6--7--8--9 24             -1
26            |  |              |
27           36 18-19-20-21-22-23             -2
28            |
29           37-38-39-40-41-...               -3
30
31                     ^
32           -3 -2 -1 X=0 1  2  3
33
34       The path is as if around a heptagon, with the left and bottom here as
35       two sides of the heptagon straightened out, and the flat top here
36       skewed across to fit a square grid.
37
38   N Start
39       The default is to number points starting N=1 as shown above.  An
40       optional "n_start" can give a different start, in the same pattern.
41       For example to start at 0,
42
43           30 29 28 27              n_start => 0
44           31 13 12 11 26
45           32 14  3  2 10 25
46           33 15  4  0  1  9 24
47           34 16  5  6  7  8 23
48           35 17 18 19 20 21 22
49           36 37 38 39 40 ...
50

FUNCTIONS

52       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
53       classes.
54
55       "$path = Math::PlanePath::HeptSpiralSkewed->new ()"
56       "$path = Math::PlanePath::HeptSpiralSkewed->new (n_start => $n)"
57           Create and return a new path object.
58
59       "$n = $path->xy_to_n ($x,$y)"
60           Return the point number for coordinates "$x,$y".  $x and $y are
61           each rounded to the nearest integer, which has the effect of
62           treating each N in the path as centred in a square of side 1, so
63           the entire plane is covered.
64

FORMULAS

66   N to X,Y
67       It's convenient to work in terms of Nstart=0 and to take each loop as
68       beginning on the South-West diagonal,
69
70                     top length = d
71
72                     30-29-28-27
73                      |         \
74                     31          26    diagonal length = d
75          left        |            \
76          length     32             25
77           = 2*d      |               \
78                     33        0       24
79                      |                 |    right
80                     34     .          23    length = d-1
81                      |                 |
82                     35 17-18-19-20-21-22
83                      |
84                      .    bottom length = 2*d-1
85
86       The SW diagonal is N=0,5,17,36,etc which is
87
88           N = (7d-11)*d/2 + 2           # starting d=1 first loop
89
90       This can be inverted to get d from N
91
92           d = floor( (sqrt(56*N+9)+11)/14 )
93
94       The side lengths are as shown above.  The first loop is d=1 and for it
95       the "right" vertical length is zero, so no such side on that first loop
96       0 <= N < 5.
97

OEIS

99       Entries in Sloane's Online Encyclopedia of Integer Sequences related to
100       this path include
101
102           <http://oeis.org/A192136> (etc)
103
104           n_start=1
105             A140065    N on Y axis
106
107           n_start=0
108             A001106    N on X axis, 9-gonal numbers
109             A218471    N on Y axis
110             A022265    N on X negative axis
111             A179986    N on Y negative axis, second 9-gonals
112             A195023    N on X=Y diagonal
113             A022264    N on North-West diagonal
114             A186029    N on South-West diagonal
115             A024966    N on South-East diagonal
116

SEE ALSO

118       Math::PlanePath, Math::PlanePath::SquareSpiral,
119       Math::PlanePath::PentSpiralSkewed, Math::PlanePath::HexSpiralSkewed
120

HOME PAGE

122       <http://user42.tuxfamily.org/math-planepath/index.html>
123

LICENSE

125       Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019,
126       2020 Kevin Ryde
127
128       This file is part of Math-PlanePath.
129
130       Math-PlanePath is free software; you can redistribute it and/or modify
131       it under the terms of the GNU General Public License as published by
132       the Free Software Foundation; either version 3, or (at your option) any
133       later version.
134
135       Math-PlanePath is distributed in the hope that it will be useful, but
136       WITHOUT ANY WARRANTY; without even the implied warranty of
137       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
138       General Public License for more details.
139
140       You should have received a copy of the GNU General Public License along
141       with Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.
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145perl v5.34.0                      2021-07-2M2ath::PlanePath::HeptSpiralSkewed(3)
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