1Math::PlanePath::CornerUAsletrerCnoanttirnigb(u3t)ed PerMlatDho:c:uPmleannteaPtaitohn::CornerAlternating(3)
2
6 Math::PlanePath::CornerAlternating -- points shaped around a corner
7 alternately
8
10 use Math::PlanePath::CornerAlternating;
11 my $path = Math::PlanePath::CornerAlternating->new;
12 my ($x, $y) = $path->n_to_xy (123);
13
15 This path is points in layers around a square outwards from a corner in
16 the first quadrant, alternately upward or downward. Each row/column
17 "gnomon" added to a square makes a one-bigger square.
18 4 | 17--18--19--20--21 ...
20 | | | |
21 3 | 16--15--14--13 22 29
22 | | | |
23 2 | 5---6---7 12 23 28
24 | | | | | |
25 1 | 4---3 8 11 24 27
26 | | | | | |
27 Y=0 | 1---2 9--10 25--26
28 +-------------------------
29 X=0 1 2 3 4 5
30 This is like the Corner path, but here gnomons go back and forward and
32 in particular so points are always a unit step apart.
33 Wider
35 An optional "wider => $integer" makes the path wider horizontally,
36 becoming a rectangle. For example
37 4 | 29--30--31--32--33--34--35--36 ...
39 | | | |
40 3 | 28--27--26--25--24--23--22 37 44 wider => 3
41 | | | |
42 2 | 11--12--13--14--15--16 21 38 43
43 | | | | | |
44 1 | 10---9---8---7---6 17 20 39 42
45 | | | | | |
46 Y=0 | 1---2---3---4---5 18--19 40--41
47 +--------------------------------------
48 X=0 1 2 3 4 5 6 7 8
49 Each gnomon has the horizontal part "wider" many steps longer. For
51 wider=3 shown, the additional points are 2,3,4 in the first row, then
52 5..10 are the next gnomon. Each gnomon is still 2 longer than the
53 previous since this widening is a constant amount in each.
54 N Start
56 The default is to number points starting N=1 as shown above. An
57 optional "n_start" can give a different start with the same shape etc.
58 For example to start at 0,
59 4 | 16 17 18 19 20
61 3 | 15 14 13 12 21 n_start => 0
62 2 | 4 5 6 11 22
63 1 | 3 2 7 10 23
64 Y=0 | 0 1 8 9 24
65 ---------------------
66 X=0 1 2 3 4
67 With Nstart=0, the pronic numbers are on the X=Y leading diagonal.
69
71 See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
72 classes.
73 "$path = Math::PlanePath::CornerAlternating->new ()"
75 "$path = Math::PlanePath::CornerAlternating->new (wider => $w, n_start
76 => $n)"
77 Create and return a new path object.
78 "($x,$y) = $path->n_to_xy ($n)"
80 Return the X,Y coordinates of point number $n on the path.
81 For "$n < n_start()" the return is an empty list. Fractional $n
83 gives an X,Y position along a straight line between the integer
84 positions.
85 "$n = $path->xy_to_n ($x,$y)"
87 Return the point number for coordinates "$x,$y".
88 $x and $y are each rounded to the nearest integer, which has the
90 effect of treating each point as a square of side 1, so the
91 quadrant x>=-0.5 and y>=-0.5 is entirely covered.
92 "($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)"
94 The returned range is exact, meaning $n_lo and $n_hi are the
95 smallest and biggest in the rectangle.
96
98 Most calculations are similar to the Corner path (without the 0.5
99 fractional part), and a reversal applied when the d gnomon number is
100 odd. When wider>0, that reversal must allow for the horizontals and
101 verticals different lengths.
102 Rectangle N Range
104 For "rect_to_n_range()", the largest gnomon is either the top or right
105 of the rectangle, depending where the top right corner x2,y2 falls
106 relative to the leading diagonal,
107 | A---B / x2<y2 | / x2>y2
109 | | |/ top | +------B right
110 | | | row | | / | side
111 | | /| biggest | | / | biggest
112 | +---+ gnomon | +------C gnomon
113 | / | /
114 +--------- +-----------
115 Then the maximum is at A or B, or B or C according as which way that
117 gnomon goes, so odd or even.
118 If it happens that B is on the diagonal, so x2=y2, then it's either A
120 or C according as the gnomon odd or even
121 | /
123 | A----+ x2=y2
124 | | /|
125 | | / |
126 | +----C
127 | /
128 +-----------
129 For wider > 0, the diagonal shifts across so that x2-wider <=> y2 is
131 the relevant test.
132
134 This path is in Sloane's Online Encyclopedia of Integer Sequences as,
135 <http://oeis.org/A319289> (etc)
137 wider=0, n_start=1 (the defaults)
139 A220603 X+1 coordinate
140 A220604 Y+1 coordinate
141 A213088 X+Y sum
142 A081346 N on X axis
143 A081345 N on Y axis
144 A002061 N on X=Y diagonal, extra initial 1
145 A081344 permutation N by diagonals
146 A194280 inverse
147 A020703 permutation N at transpose Y,X
148 A027709 boundary length of N unit squares
150 A078633 grid sticks of N points
151 n_start=0
153 A319290 X coordinate
154 A319289 Y coordinate
155 A319514 Y,X coordinate pairs
156 A329116 X-Y diff
157 A053615 abs(X-Y) diff
158 A000196 max(X,Y), being floor(sqrt(N))
159 A339265 dX-dY increments (runs +1,-1)
160 A002378 N on X=Y diagonal, pronic numbers
161 A220516 permutation N by diagonals
162 n_start=2
164 A014206 N on X=Y diagonal, pronic+2
165 wider=1, n_start=1
167 A081347 N on X axis
168 A081348 N on Y axis
169 A080335 N on X=Y diagonal
170 A093650 permutation N by diagonals
171 wider=1, n_start=0
173 A180714 X-Y diff
174 wider=2, n_start=1
176 A081350 N on X axis
177 A081351 N on Y axis
178 A081352 N on X=Y diagonal
179 A081349 permutation N by diagonals
180
182 Math::PlanePath, Math::PlanePath::Corner,
183 Math::PlanePath::DiagonalsAlternating
184
186 <http://user42.tuxfamily.org/math-planepath/index.html>
187
189 Copyright 2021 Kevin Ryde
190 This file is part of Math-PlanePath.
192 Math-PlanePath is free software; you can redistribute it and/or modify
194 it under the terms of the GNU General Public License as published by
195 the Free Software Foundation; either version 3, or (at your option) any
196 later version.
197 Math-PlanePath is distributed in the hope that it will be useful, but
199 WITHOUT ANY WARRANTY; without even the implied warranty of
200 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
201 General Public License for more details.
202 You should have received a copy of the GNU General Public License along
204 with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
205perl v5.34.0 2021-07-M2a2th::PlanePath::CornerAlternating(3)