1Math::PlanePath::Hypot(U3s)er Contributed Perl DocumentatMiaotnh::PlanePath::Hypot(3)
2
6 Math::PlanePath::Hypot -- points in order of hypotenuse distance
7
9 use Math::PlanePath::Hypot;
10 my $path = Math::PlanePath::Hypot->new;
11 my ($x, $y) = $path->n_to_xy (123);
12
14 This path visits integer points X,Y in order of their distance from the
15 origin 0,0, or anti-clockwise from the X axis among those of equal
16 distance,
17 84 73 83 5
19 74 64 52 47 51 63 72 4
20 75 59 40 32 27 31 39 58 71 3
21 65 41 23 16 11 15 22 38 62 2
22 85 53 33 17 7 3 6 14 30 50 82 1
23 76 48 28 12 4 1 2 10 26 46 70 <- Y=0
24 86 54 34 18 8 5 9 21 37 57 89 -1
25 66 42 24 19 13 20 25 45 69 -2
26 77 60 43 35 29 36 44 61 81 -3
27 78 67 55 49 56 68 80 -4
28 87 79 88 -5
29 ^
31 -5 -4 -3 -2 -1 X=0 1 2 3 4 5
32 For example N=58 is at X=4,Y=-1 is sqrt(4*4+1*1) = sqrt(17) from the
34 origin. The next furthest from the origin is X=3,Y=3 at sqrt(18).
35 See "TriangularHypot" for points in order of X^2+3*Y^2, or
37 "DiamondSpiral" and "PyrmaidSides" in order of plain sum X+Y.
38 Equal Distances
40 Points with the same distance are taken in anti-clockwise order around
41 from the X axis. For example X=3,Y=1 is sqrt(10) from the origin, as
42 are the swapped X=1,Y=3, and X=-1,Y=3 etc in other quadrants, for a
43 total 8 points N=30 to N=37 all the same distance.
44 When one of X or Y is 0 there's no negative, so just four negations
46 like N=10 to 13 points X=2,Y=0 through X=0,Y=-2. Or on the diagonal
47 X==Y there's no swap, so just four like N=22 to N=25 points X=3,Y=3
48 through X=3,Y=-3.
49 There can be more than one way for the same distance to arise. A
51 Pythagorean triple like 3^2 + 4^2 == 5^2 has 8 points from the 3,4,
52 then 4 points from the 5,0 giving a total 12 points N=70 to N=81.
53 Other combinations like 20^2 + 15^2 == 24^2 + 7^2 occur too, and also
54 with more than two different ways to have the same sum.
55 Multiples of 4
57 The first point of a given distance from the origin is either on the X
58 axis or somewhere in the first octant. The row Y=1 just above the axis
59 is the first of its equals from X>=2 onwards, and similarly further
60 rows for big enough X.
61 There's always a multiple of 4 many points with the same distance so
63 the first point has N=4*k+2, and similarly on the negative X side
64 N=4*j, for some k or j. If you plot the prime numbers on the path then
65 those even N's (composites) are gaps just above the positive X axis,
66 and on or just below the negative X axis.
67 Circle Lattice
69 Gauss's circle lattice problem asks how many integer X,Y points there
70 are within a circle of radius R.
71 The points on the X axis N=2,10,26,46, etc are the first for which
73 X^2+Y^2==R^2 (integer X==R). Adding option "n_start=>0" to make them
74 each 1 less gives the number of points strictly inside, ie. X^2+Y^2 <
75 R^2.
76 The last point satisfying X^2+Y^2==R^2 is either in the octant below
78 the X axis, or is on the negative Y axis. Those N's are the number of
79 points X^2+Y^2<=R^2, Sloane's A000328.
80 When that A000328 sequence is plotted on the path a straight line can
82 be seen in the fourth quadrant extending down just above the diagonal.
83 It arises from multiples of the Pythagorean 3^2 + 4^2, first X=4,Y=-3,
84 then X=8,Y=-6, etc X=4*k,Y=-3*k. But sometimes the multiple is not the
85 last among those of that 5*k radius, so there's gaps in the line. For
86 example 20,-15 is not the last since because 24,-7 is also 25 away from
87 the origin.
88 Even Points
90 Option "points => "even"" visits just the even points, meaning the sum
91 X+Y even, so X,Y both even or both odd.
92 points => "even"
94 52 40 39 51 5
96 47 32 23 31 46 4
97 53 27 16 15 26 50 3
98 33 11 7 10 30 2
99 41 17 3 2 14 38 1
100 24 8 1 6 22 <- Y=0
101 42 18 4 5 21 45 -1
102 34 12 9 13 37 -2
103 54 28 19 20 29 57 -3
104 48 35 25 36 49 -4
105 55 43 44 56 -5
106 ^
108 -5 -4 -3 -2 -1 X=0 1 2 3 4 5
109 Even points can be mapped to all points by a 45 degree rotate and flip.
111 N=1,6,22,etc on the X axis here is on the X=Y diagonal of all-points.
112 And conversely N=1,2,10,26,etc on the X=Y diagonal here is the X axis
113 of all-points.
114 The sets of points with equal hypotenuse are the same in the even and
116 all, but the flip takes them in a reversed order.
117 Odd Points
119 Option "points => "odd"" visits just the odd points, meaning sum X+Y
120 odd, so X,Y one odd the other even.
121 points => "odd"
123 71 55 54 70 6
126 63 47 36 46 62 5
127 64 37 27 26 35 61 4
128 72 38 19 14 18 34 69 3
129 48 20 7 6 17 45 2
130 56 28 8 2 5 25 53 1
131 39 15 3 + 1 13 33 <- Y=0
132 57 29 9 4 12 32 60 -1
133 49 21 10 11 24 52 -2
134 73 40 22 16 23 44 76 -3
135 65 41 30 31 43 68 -4
136 66 50 42 51 67 -5
137 74 58 59 75 -6
138 ^
140 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6
141 Odd points can be mapped to all points by a 45 degree rotate and a
143 shift X-1,Y+1 to put N=1 at the origin. The effect of that shift is as
144 if the hypot measure in "all" points was (X-1/2)^2+(Y-1/2)^2 and for
145 that reason the sets of points with equal hypots are not the same in
146 odd and all.
147
149 See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
150 classes.
151 "$path = Math::PlanePath::Hypot->new ()"
153 "$path = Math::PlanePath::Hypot->new (points => $str), n_start => $n"
154 Create and return a new hypot path object. The "points" option can
155 be
156 "all" all integer X,Y (the default)
158 "even" only points with X+Y even
159 "odd" only points with X+Y odd
160 "($x,$y) = $path->n_to_xy ($n)"
162 Return the X,Y coordinates of point number $n on the path.
163 For "$n < 1" the return is an empty list, it being considered the
165 first point at X=0,Y=0 is N=1.
166 Currently it's unspecified what happens if $n is not an integer.
168 Successive points are a fair way apart, so it may not make much
169 sense to say give an X,Y position in between the integer $n.
170 "$n = $path->xy_to_n ($x,$y)"
172 Return an integer point number for coordinates "$x,$y". Each
173 integer N is considered the centre of a unit square and an "$x,$y"
174 within that square returns N.
175 For "even" and "odd" options only every second square in the plane
177 has an N and if "$x,$y" is a position not covered then the return
178 is "undef".
179
181 The calculations are not particularly efficient currently. Private
182 arrays are built similar to what's described for "HypotOctant", but
183 with replication for negative and swapped X,Y.
184
186 Entries in Sloane's Online Encyclopedia of Integer Sequences related to
187 this path include
188 <http://oeis.org/A051132> (etc)
190 points="all", n_start=0
192 A051132 N on X axis, being count points norm < X^2
193 points="odd"
195 A005883 count of points with norm==4*n+1
196
198 Math::PlanePath, Math::PlanePath::HypotOctant,
199 Math::PlanePath::TriangularHypot, Math::PlanePath::PixelRings,
200 Math::PlanePath::PythagoreanTree
201
203 <http://user42.tuxfamily.org/math-planepath/index.html>
204
206 Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020
207 Kevin Ryde
208 This file is part of Math-PlanePath.
210 Math-PlanePath is free software; you can redistribute it and/or modify
212 it under the terms of the GNU General Public License as published by
213 the Free Software Foundation; either version 3, or (at your option) any
214 later version.
215 Math-PlanePath is distributed in the hope that it will be useful, but
217 WITHOUT ANY WARRANTY; without even the implied warranty of
218 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
219 General Public License for more details.
220 You should have received a copy of the GNU General Public License along
222 with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
223perl v5.38.0 2023-07-20 Math::PlanePath::Hypot(3)