1Math::PlanePath::DiagonUaslesrAlCtoenrtnraitbiuntge(d3M)Paetrhl::DPolcaunmeePnattaht:i:oDniagonalsAlternating(3)
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6 Math::PlanePath::DiagonalsAlternating -- points in diagonal stripes of
7 alternating directions
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10 use Math::PlanePath::DiagonalsAlternating;
11 my $path = Math::PlanePath::DiagonalsAlternating->new;
12 my ($x, $y) = $path->n_to_xy (123);
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15 This path follows successive diagonals going from the Y axis down to
16 the X axis and then back again,
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18 7 | 29
19 6 | 28 30
20 5 | 16 27 31
21 4 | 15 17 26 ...
22 3 | 7 14 18 25
23 2 | 6 8 13 19 24
24 1 | 2 5 9 12 20 23
25 Y=0 | 1 3 4 10 11 21 22
26 +----------------------------
27 X=0 1 2 3 4 5 6
28
29 The triangular numbers 1,3,6,10,etc k*(k+1)/2 are the start of each run
30 up or down alternately on the X axis and Y axis. N=1,6,15,28,etc on
31 the Y axis (Y even) are the hexagonal numbers j*(2j-1).
32 N=3,10,21,36,etc on the X axis (X odd) are the hexagonal numbers of the
33 second kind j*(2j+1).
34
35 N Start
36 The default is to number points starting N=1 as shown above. An
37 optional "n_start" can give a different start, in the same pattern.
38 For example to start at 0,
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40 n_start => 0
41
42 4 | 14
43 3 | 6 13
44 2 | 5 7 12
45 1 | 1 4 8 11
46 Y=0 | 0 2 3 9 10
47 +-----------------
48 X=0 1 2 3 4
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51 See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
52 classes.
53
54 "$path = Math::PlanePath::DiagonalsAlternating->new ()"
55 "$path = Math::PlanePath::DiagonalsAlternating->new (n_start => $n)"
56 Create and return a new path object.
57
58 "($x,$y) = $path->n_to_xy ($n)"
59 Return the X,Y coordinates of point number $n on the path.
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61 For "$n < 1" the return is an empty list, it being considered the
62 path begins at 1.
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65 Rectangle to N Range
66 Within each row increasing X is increasing N, and in each column
67 increasing Y is increasing N. So in a rectangle the lower left corner
68 is the minimum N and the upper right is the maximum N.
69
70 | N max
71 | ----------+
72 | | ^ |
73 | | | |
74 | | ----> |
75 | +----------
76 | N min
77 +-------------------
78
80 Entries in Sloane's Online Encyclopedia of Integer Sequences related to
81 this path include
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83 <http://oeis.org/A131179> (etc)
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85 n_start=1
86 A131179 N on X axis (extra initial 0)
87 A128918 N on Y axis (extra initial 1)
88 A001844 N on X=Y diagonal
89 A038722 permutation N at transpose Y,X
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91 n_start=0
92 A003056 X+Y
93 A004247 X*Y
94 A049581 abs(X-Y)
95 A048147 X^2+Y^2
96 A004198 X bit-and Y
97 A003986 X bit-or Y
98 A003987 X bit-xor Y
99 A004197 min(X,Y)
100 A003984 max(X,Y)
101 A101080 HammingDist(X,Y)
102 A023531 dSum = dX+dY, being 1 at N=triangular+1 (and 0)
103 A046092 N on X=Y diagonal
104 A061579 permutation N at transpose Y,X
105
106 A056011 permutation N at points by Diagonals,direction=up order
107 A056023 permutation N at points by Diagonals,direction=down
108 runs alternately up and down, both are self-inverse
109
110 The coordinates such as A003056 X+Y are the same here as in the
111 Diagonals path. "DiagonalsAlternating" transposes X,Y -> Y,X in every
112 second diagonal but forms such as X+Y are unchanged by swapping to Y+X.
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115 Math::PlanePath, Math::PlanePath::Diagonals,
116 Math::PlanePath::DiagonalsOctant
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119 <http://user42.tuxfamily.org/math-planepath/index.html>
120
122 Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018 Kevin
123 Ryde
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125 This file is part of Math-PlanePath.
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127 Math-PlanePath is free software; you can redistribute it and/or modify
128 it under the terms of the GNU General Public License as published by
129 the Free Software Foundation; either version 3, or (at your option) any
130 later version.
131
132 Math-PlanePath is distributed in the hope that it will be useful, but
133 WITHOUT ANY WARRANTY; without even the implied warranty of
134 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
135 General Public License for more details.
136
137 You should have received a copy of the GNU General Public License along
138 with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
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142perl v5.30.1 2020-M0a1t-h3:0:PlanePath::DiagonalsAlternating(3)