1Math::PlanePath::DiagonUaslesr(3C)ontributed Perl DocumeMnattaht:i:oPnlanePath::Diagonals(3)
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6 Math::PlanePath::Diagonals -- points in diagonal stripes
7
9 use Math::PlanePath::Diagonals;
10 my $path = Math::PlanePath::Diagonals->new;
11 my ($x, $y) = $path->n_to_xy (123);
12
14 This path follows successive diagonals going from the Y axis down to
15 the X axis.
16 6 | 22
18 5 | 16 23
19 4 | 11 17 24
20 3 | 7 12 18 ...
21 2 | 4 8 13 19
22 1 | 2 5 9 14 20
23 Y=0 | 1 3 6 10 15 21
24 +-------------------------
25 X=0 1 2 3 4 5
26 N=1,3,6,10,etc on the X axis is the triangular numbers.
28 N=1,2,4,7,11,etc on the Y axis is the triangular plus 1, the next point
29 visited after the X axis.
30 Direction
32 Option "direction => 'up'" reverses the order within each diagonal to
33 count upward from the X axis.
34 direction => "up"
36 5 | 21
38 4 | 15 20
39 3 | 10 14 19 ...
40 2 | 6 9 13 18 24
41 1 | 3 5 8 12 17 23
42 Y=0 | 1 2 4 7 11 16 22
43 +-----------------------------
44 X=0 1 2 3 4 5 6
45 This is merely a transpose changing X,Y to Y,X, but it's the same as in
47 "DiagonalsOctant" and can be handy to control the direction when
48 combining "Diagonals" with some other path or calculation.
49 N Start
51 The default is to number points starting N=1 as shown above. An
52 optional "n_start" can give a different start, in the same diagonals
53 sequence. For example to start at 0,
54 n_start => 0, n_start=>0
56 direction=>"down" direction=>"up"
57 4 | 10 | 14
59 3 | 6 11 | 9 13
60 2 | 3 7 12 | 5 8 12
61 1 | 1 4 8 13 | 2 4 7 11
62 Y=0 | 0 2 5 9 14 | 0 1 3 6 10
63 +----------------- +-----------------
64 X=0 1 2 3 4 X=0 1 2 3 4
65 N=0,1,3,6,10,etc on the Y axis of "down" or the X axis of "up" is the
67 triangular numbers Y*(Y+1)/2.
68 X,Y Start
70 Options "x_start => $x" and "y_start => $y" give a starting position
71 for the diagonals. For example to start at X=1,Y=1
72 7 | 22 x_start => 1,
74 6 | 16 23 y_start => 1
75 5 | 11 17 24
76 4 | 7 12 18 ...
77 3 | 4 8 13 19
78 2 | 2 5 9 14 20
79 1 | 1 3 6 10 15 21
80 Y=0 |
81 +------------------
82 X=0 1 2 3 4 5
83 The effect is merely to add a fixed offset to all X,Y values taken and
85 returned, but it can be handy to have the path do that to step through
86 non-negatives or similar.
87
89 See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
90 classes.
91 "$path = Math::PlanePath::Diagonals->new ()"
93 "$path = Math::PlanePath::Diagonals->new (direction => $str, n_start =>
94 $n, x_start => $x, y_start => $y)"
95 Create and return a new path object. The "direction" option (a
96 string) can be
97 direction => "down" the default
99 direction => "up" number upwards from the X axis
100 "($x,$y) = $path->n_to_xy ($n)"
102 Return the X,Y coordinates of point number $n on the path.
103 For "$n < 0.5" the return is an empty list, it being considered the
105 path begins at 1.
106 "$n = $path->xy_to_n ($x,$y)"
108 Return the point number for coordinates "$x,$y". $x and $y are
109 each rounded to the nearest integer, which has the effect of
110 treating each point $n as a square of side 1, so the quadrant
111 x>=-0.5, y>=-0.5 is entirely covered.
112 "($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)"
114 The returned range is exact, meaning $n_lo and $n_hi are the
115 smallest and biggest in the rectangle.
116
118 X,Y to N
119 The sum d=X+Y numbers each diagonal from d=0 upwards, corresponding to
120 the Y coordinate where the diagonal starts (or X if direction=up).
121 d=2
123 \
124 d=1 \
125 \ \
126 d=0 \ \
127 \ \ \
128 N is then given by
130 d = X+Y
132 N = d*(d+1)/2 + X + Nstart
133 The d*(d+1)/2 shows how the triangular numbers fall on the Y axis when
135 X=0 and Nstart=0. For the default Nstart=1 it's 1 more than the
136 triangulars, as noted above.
137 d can be expanded out to the following quite symmetric form. This
139 almost suggests something parabolic but is still the straight line
140 diagonals.
141 X^2 + 3X + 2XY + Y + Y^2
143 N = ------------------------ + Nstart
144 2
145 N to X,Y
147 The above formula N=d*(d+1)/2 can be solved for d as
148 d = floor( (sqrt(8*N+1) - 1)/2 )
150 # with n_start=0
151 For example N=12 is d=floor((sqrt(8*12+1)-1)/2)=4 as that N falls in
153 the fifth diagonal. Then the offset from the Y axis NY=d*(d-1)/2 is
154 the X position,
155 X = N - d*(d-1)/2
157 Y = d - X
158 In the code fractional N is handled by imagining each diagonal
160 beginning 0.5 back from the Y axis. That's handled by adding 0.5 into
161 the sqrt, which is +4 onto the 8*N.
162 d = floor( (sqrt(8*N+5) - 1)/2 )
164 # N>=-0.5
165 The X and Y formulas are unchanged, since N=d*(d-1)/2 is still the Y
167 axis. But each diagonal d begins up to 0.5 before that and therefor X
168 extends back to -0.5.
169 Rectangle to N Range
171 Within each row increasing X is increasing N, and in each column
172 increasing Y is increasing N. So in a rectangle the lower left corner
173 is minimum N and the upper right is maximum N.
174 | \ \ N max
176 | \ ----------+
177 | | \ |\
178 | |\ \ |
179 | \| \ \ |
180 | +----------
181 | N min \ \ \
182 +-------------------------
183
185 Entries in Sloane's Online Encyclopedia of Integer Sequences related to
186 this path include
187 <http://oeis.org/A002262> (etc)
189 direction=down (the default)
191 A002262 X coordinate, runs 0 to k
192 A025581 Y coordinate, runs k to 0
193 A003056 X+Y coordinate sum, k repeated k+1 times
194 A114327 Y-X coordinate diff
195 A101080 HammingDist(X,Y)
196 A127949 dY, change in Y coordinate
198 A000124 N on Y axis, triangular numbers + 1
200 A001844 N on X=Y diagonal
201 A185787 total N in row to X=Y diagonal
203 A185788 total N in row to X=Y-1
204 A100182 total N in column to Y=X diagonal
205 A101165 total N in column to Y=X-1
206 A185506 total N in rectangle 0,0 to X,Y
207 either direction=up,down
209 A097806 turn 0=straight, 1=not straight
210 direction=down, x_start=1, y_start=1
212 A057555 X,Y pairs
213 A057046 X at N=2^k
214 A057047 Y at N=2^k
215 direction=down, n_start=0
217 A057554 X,Y pairs
218 A023531 dSum = dX+dY, being 1 at N=triangular+1 (and 0)
219 A000096 N on X axis, X*(X+3)/2
220 A000217 N on Y axis, the triangular numbers
221 A129184 turn 1=left,0=right
222 A103451 turn 1=left or right,0=straight, but extra initial 1
223 A103452 turn 1=left,0=straight,-1=right, but extra initial 1
224 direction=up, n_start=0
225 A129184 turn 0=left,1=right
226 direction=up, n_start=-1
228 A023531 turn 1=left,0=right
229 direction=down, n_start=-1
230 A023531 turn 0=left,1=right
231 in direction=up the X,Y coordinate forms are the same but swap X,Y
233 either direction=up,down
235 A038722 permutation N at transpose Y,X
236 which is direction=down <-> direction=up
237 either direction, x_start=1, y_start=1
239 A003991 X*Y coordinate product
240 A003989 GCD(X,Y) greatest common divisor starting (1,1)
241 A003983 min(X,Y)
242 A051125 max(X,Y)
243 either direction, n_start=0
245 A049581 abs(X-Y) coordinate diff
246 A004197 min(X,Y)
247 A003984 max(X,Y)
248 A004247 X*Y coordinate product
249 A048147 X^2+Y^2
250 A109004 GCD(X,Y) greatest common divisor starting (0,0)
251 A004198 X bit-and Y
252 A003986 X bit-or Y
253 A003987 X bit-xor Y
254 A156319 turn 0=straight,1=left,2=right
255 A061579 permutation N at transpose Y,X
257 which is direction=down <-> direction=up
258
260 Math::PlanePath, Math::PlanePath::DiagonalsAlternating,
261 Math::PlanePath::DiagonalsOctant, Math::PlanePath::Corner,
262 Math::PlanePath::Rows, Math::PlanePath::Columns
263
265 <http://user42.tuxfamily.org/math-planepath/index.html>
266
268 Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019
269 Kevin Ryde
270 This file is part of Math-PlanePath.
272 Math-PlanePath is free software; you can redistribute it and/or modify
274 it under the terms of the GNU General Public License as published by
275 the Free Software Foundation; either version 3, or (at your option) any
276 later version.
277 Math-PlanePath is distributed in the hope that it will be useful, but
279 WITHOUT ANY WARRANTY; without even the implied warranty of
280 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
281 General Public License for more details.
282 You should have received a copy of the GNU General Public License along
284 with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
285perl v5.32.0 2020-07-28 Math::PlanePath::Diagonals(3)