1Math::PlanePath::TriangUuslearrHCyopnottr(i3b)uted PerlMDaotchu:m:ePnltaanteiPoanth::TriangularHypot(3)
2
6 Math::PlanePath::TriangularHypot -- points of triangular lattice in
7 order of hypotenuse distance
8
10 use Math::PlanePath::TriangularHypot;
11 my $path = Math::PlanePath::TriangularHypot->new;
12 my ($x, $y) = $path->n_to_xy (123);
13
15 This path visits X,Y points on a triangular "A2" lattice in order of
16 their distance from the origin 0,0 and anti-clockwise around from the X
17 axis among those of equal distance.
18 58 47 39 46 57 4
20 48 34 23 22 33 45 3
22 40 24 16 9 15 21 38 2
24 49 25 10 4 3 8 20 44 1
26 35 17 5 1 2 14 32 <- Y=0
28 50 26 11 6 7 13 31 55 -1
30 41 27 18 12 19 30 43 -2
32 51 36 28 29 37 54 -3
34 60 52 42 53 61 -4
36 ^
38 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7
39 The lattice is put on a square X,Y grid using every second point per
41 "Triangular Lattice" in Math::PlanePath. Scaling X/2,Y*sqrt(3)/2 gives
42 equilateral triangles of side length 1 making a distance from X,Y to
43 the origin
44 dist^2 = (X/2^2 + (Y*sqrt(3)/2)^2
46 = (X^2 + 3*Y^2) / 4
47 For example N=19 at X=2,Y=-2 is sqrt((2**2+3*-2**2)/4) = sqrt(4) from
49 the origin. The next smallest after that is X=5,Y=1 at sqrt(7). The
50 key part is X^2 + 3*Y^2 as the distance measure to order the points.
51 Equal Distances
53 Points with the same distance are taken in anti-clockwise order around
54 from the X axis. For example N=14 at X=4,Y=0 is sqrt(4) from the
55 origin, and so are the rotated X=2,Y=2 and X=-2,Y=2 etc in other sixth
56 segments, for a total 6 points N=14 to N=19 all the same distance.
57 Symmetry means there's a set of 6 or 12 points with the same distance,
59 so the count of same-distance points is always a multiple of 6 or 12.
60 There are 6 symmetric points when on the six radial lines X=0, X=Y or
61 X=-Y, and on the lines Y=0, X=3*Y or X=-3*Y which are midway between
62 them. There's 12 symmetric points for anything else, ie. anything in
63 the twelve slices between those twelve lines. The first set of 12
64 equal is N=20 to N=31 all at sqrt(28).
65 There can also be further ways for the same distance to arise, as
67 multiple solutions to X^2+3*Y^3=d^2, but the 6-way or 12-way symmetry
68 means there's always a multiple of 6 or 12 in total.
69 Odd Points
71 Option "points => "odd"" visits just the odd points, meaning sum X+Y
72 odd, which is X,Y one odd the other even.
73 points => "odd"
75 69 5
76 66 50 45 44 49 65 4
77 58 40 28 25 27 39 57 3
78 54 32 20 12 11 19 31 53 2
79 36 16 6 3 5 15 35 1
80 46 24 10 2 1 9 23 43 <- Y=0
81 37 17 7 4 8 18 38 -1
82 55 33 21 13 14 22 34 56 -2
83 59 41 29 26 30 42 60 -3
84 67 51 47 48 52 68 -4
85 70 -5
86 ^
88 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6
89 All Points
91 Option "points => "all"" visits all integer X,Y points.
92 points => "all"
94 64 59 49 44 48 58 63 3
96 69 50 39 30 25 19 24 29 38 47 68 2
97 51 35 20 13 8 4 7 12 18 34 46 1
98 65 43 31 17 9 3 1 2 6 16 28 42 62 <- Y=0
99 52 36 21 14 10 5 11 15 23 37 57 -1
100 70 53 40 32 26 22 27 33 41 56 71 -2
101 66 60 54 45 55 61 67 -3
102 ^
104 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6
105 Hex Points
107 Option "points => "hex"" visits X,Y points making a hexagonal grid,
108 points => "hex"
110 50----42 49----59 5
112 / \ / \
113 51----39 27----33 48 4
114 / \ / \ /
115 43 22----15 21----32 3
116 \ / \ / \
117 28----16 6----11 26----41 2
118 / \ / \ / \
119 52----34 7---- 3 5----14 47 1
120 / \ / \ / \ /
121 60 23----12 1-----2 20----38 <- Y=0
122 \ / \ / \ / \
123 53----35 8---- 4 10----19 58 -1
124 \ / \ / \ /
125 29----17 9----13 31----46 -2
126 / \ / \ /
127 44 24----18 25----37 -3
128 \ / \ / \
129 54----40 30----36 57 -4
130 \ / \ /
131 55----45 56----61 -5
132 ^
134 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8 9
135 N=1 is at the origin X=0,Y=0, then N=2,3,4 are all at X^2+3Y^2=4 away
137 from the origin, etc. The joining lines drawn above show the grid
138 pattern but points are in order of distance from the origin.
139 The points are all integer X,Y with X+3Y mod 6 == 0 or 2. This is a
141 subset of the default "even" points in that X+Y is even but with 1 of
142 each 3 points skipped to make the hexagonal outline.
143 Hex Rotated Points
145 Option "points => "hex_rotated"" is the same hexagonal points but
146 rotated around so N=2 is at +60 degrees instead of on the X axis.
147 points => "hex_rotated"
149 60----50 42----49 5
152 / \ / \
153 51 33----27 38----48 4
154 \ / \ / \
155 34----22 15----21 41 3
156 / \ / \ /
157 43----28 12-----6 14----26 2
158 / \ / \ / \
159 52 16-----7 2-----5 32----47 1
160 \ / \ / \ / \
161 39----23 3-----1 11----20 59 <- Y=0
162 / \ / \ / \ /
163 53 17-----8 4----10 37----58 -1
164 \ / \ / \ /
165 44----29 13-----9 19----31 -2
166 \ / \ / \
167 35----24 18----25 46 -3
168 / \ / \ /
169 54 36----30 40----57 -4
170 \ / \ /
171 61----55 45----56 -5
172 ^
175 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8 9
176 Points are still numbered from the X axis clockwise. The sets of
178 points at equal hypotenuse distances are the same as plain "hex" but
179 the numbering is changed by the rotation.
180 The points visited are all integer X,Y with X+3Y mod 6 == 0 or 4. This
182 grid can be viewed either as a +60 degree or a +180 degree rotation of
183 the plain hex.
184 Hex Centred Points
186 Option "points => "hex_centred"" is the same hexagonal grid as hex
187 above, but with the origin X=0,Y=0 in the centre of a hexagon,
188 points => "hex_centred"
190 46----45 5
192 / \
193 39----28 27----38 4
194 / \ / \
195 47----29 16----15 26----44 3
196 / \ / \ / \
197 48 17-----9 8----14 43 2
198 \ / \ / \ /
199 30----18 3-----2 13----25 1
200 / \ / \ / \
201 40 10-----4 . 1-----7 37 <- Y=0
202 \ / \ / \ /
203 31----19 5-----6 24----36 -1
204 / \ / \ / \
205 49 20----11 12----23 54 -2
206 \ / \ / \ /
207 50----32 21----22 35----53 -3
208 \ / \ /
209 41----33 34----42 -4
210 \ /
211 51----52 -5
212 ^
214 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8 9
215 N=1,2,3,4,5,6 are all at X^2+3Y^2=4 away from the origin, then
217 N=7,8,9,10,11,12, etc. The points visited are all integer X,Y with
218 X+3Y mod 6 == 2 or 4.
219
221 See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
222 classes.
223 "$path = Math::PlanePath::TriangularHypot->new ()"
225 "$path = Math::PlanePath::TriangularHypot->new (points => $str)"
226 Create and return a new hypot path object. The "points" option can
227 be
228 "even" only points with X+Y even (the default)
230 "odd" only points with X+Y odd
231 "all" all integer X,Y
232 "hex" hexagonal X+3Y==0,2 mod 6
233 "hex_rotated" hexagonal X+3Y==0,4 mod 6
234 "hex_centred" hexagonal X+3Y==2,4 mod 6
235 Create and return a new triangular hypot path object.
237 "($x,$y) = $path->n_to_xy ($n)"
239 Return the X,Y coordinates of point number $n on the path.
240 For "$n < 1" the return is an empty list as the first point at
242 X=0,Y=0 is N=1.
243 Currently it's unspecified what happens if $n is not an integer.
245 Successive points are a fair way apart, so it may not make much
246 sense to say give an X,Y position in between the integer $n.
247 "$n = $path->xy_to_n ($x,$y)"
249 Return an integer point number for coordinates "$x,$y". Each
250 integer N is considered the centre of a unit square and an "$x,$y"
251 within that square returns N.
252 For "even" and "odd" options only every second square in the plane
254 has an N and if "$x,$y" is a position not covered then the return
255 is "undef".
256
258 Entries in Sloane's Online Encyclopedia of Integer Sequences related to
259 this path include,
260 <http://oeis.org/A003136> (etc)
262 points="even" (the default)
264 A003136 norms (X^2+3*Y^2)/4 which occur
265 A004016 count of points of norm==n
266 A035019 skipping zero counts
267 A088534 counting only in the twelfth 0<=X<=Y
268 The counts in these sequences are expressed as norm = x^2+x*y+y^2.
270 That x,y is related to the "even" X,Y on the path here by a -45 degree
271 rotation,
272 x = (Y-X)/2 X = 2*(x+y)
274 y = (X+Y)/2 Y = 2*(y-x)
275 norm = x^2+x*y+y^2
277 = ((Y-X)/2)^2 + (Y-X)/2 * (X+Y)/2 + ((X+Y)/2)^2
278 = (X^2 + 3*Y^2) / 4
279 The X^2+3*Y^2 is the dist^2 described above for equilateral triangles
281 of unit side. The factor of /4 scales the distance but of course
282 doesn't change the sets of points of the same distance.
283 points="all"
285 A092572 norms X^2+3*Y^2 which occur
286 A158937 norms X^2+3*Y^2 which occur, X>0,Y>0 with repeats
287 A092573 count of points norm==n for X>0,Y>0
288 A092574 norms X^2+3*Y^2 which occur for X>0,Y>0, gcd(X,Y)=1
290 A092575 count of points norm==n for X>0,Y>0, gcd(X,Y)=1
291 ie. X,Y no common factor
292 points="hex"
294 A113062 count of points norm=X^2+3*Y^2=4*n (theta series)
295 A113063 divided by 3
296 points="hex_centred"
298 A217219 count of points norm=X^2+3*Y^2=4*n (theta series)
299
301 Math::PlanePath, Math::PlanePath::Hypot, Math::PlanePath::HypotOctant,
302 Math::PlanePath::PixelRings, Math::PlanePath::HexSpiral
303
305 <http://user42.tuxfamily.org/math-planepath/index.html>
306
308 Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019,
309 2020 Kevin Ryde
310 This file is part of Math-PlanePath.
312 Math-PlanePath is free software; you can redistribute it and/or modify
314 it under the terms of the GNU General Public License as published by
315 the Free Software Foundation; either version 3, or (at your option) any
316 later version.
317 Math-PlanePath is distributed in the hope that it will be useful, but
319 WITHOUT ANY WARRANTY; without even the implied warranty of
320 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
321 General Public License for more details.
322 You should have received a copy of the GNU General Public License along
324 with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
325perl v5.36.0 2022-07-22Math::PlanePath::TriangularHypot(3)