1Math::PlanePath::HypotOUcstearntC(o3n)tributed Perl DocuMmaetnht:a:tPiloannePath::HypotOctant(3)
2
6 Math::PlanePath::HypotOctant -- octant of points in order of hypotenuse
7 distance
8
10 use Math::PlanePath::HypotOctant;
11 my $path = Math::PlanePath::HypotOctant->new;
12 my ($x, $y) = $path->n_to_xy (123);
13
15 This path visits an octant of integer points X,Y in order of their
16 distance from the origin 0,0. The points are a rising triangle
17 0<=Y<=X,
18 8 | 61
20 7 | 47 54
21 6 | 36 43 49
22 5 | 27 31 38 44
23 4 | 18 23 28 34 39
24 3 | 12 15 19 24 30 37
25 2 | 6 9 13 17 22 29 35
26 1 | 3 5 8 11 16 21 26 33
27 Y=0 | 1 2 4 7 10 14 20 25 32 ...
28 +---------------------------------------
29 X=0 1 2 3 4 5 6 7 8
30 For example N=11 at X=4,Y=1 is sqrt(4*4+1*1) = sqrt(17) from the
32 origin. The next furthest from the origin is X=3,Y=3 at sqrt(18).
33 This octant is "primitive" elements X^2+Y^2 in the sense that it
35 excludes negative X or Y or swapped Y,X.
36 Equal Distances
38 Points with the same distance from the origin are taken in anti-
39 clockwise order from the X axis, which means by increasing Y. Points
40 with the same distance occur when there's more than one way to express
41 a given distance as the sum of two squares.
42 Pythagorean triples give a point on the X axis and also above. For
44 example 5^2 == 4^2 + 3^2 has N=14 at X=5,Y=0 simply as 5^2 = 5^2 + 0
45 and then N=15 at X=4,Y=3 for the triple. Both are 5 away from the
46 origin.
47 Combinations like 20^2 + 15^2 == 24^2 + 7^2 occur too, and also with
49 three or more different ways to have the same sum distance.
50 Even Points
52 Option "points => "even"" visits just the even points, meaning the sum
53 X+Y even, so X,Y both even or both odd.
54 12 | 66
56 11 | points => "even" 57
57 10 | 49 58
58 9 | 40 50
59 8 | 32 41 51
60 7 | 25 34 43
61 6 | 20 27 35 45
62 5 | 15 21 29 37
63 4 | 10 16 22 30 39
64 3 | 7 11 17 24 33
65 2 | 4 8 13 19 28 38
66 1 | 2 5 9 14 23 31
67 Y=0 | 1 3 6 12 18 26 36
68 +---------------------------------------
69 X=0 1 2 3 4 5 6 7 8 9 10 11 12
70 Even points can be mapped to all points by a 45 degree rotate and flip.
72 N=1,3,6,12,etc on the X axis here is on the X=Y diagonal of all-points.
73 And conversely N=1,2,4,7,10,etc on the X=Y diagonal here is on the X
74 axis of all-points.
75 all_X = (even_X + even_Y) / 2
77 all_Y = (even_X - even_Y) / 2
78 even_X = (all_X + all_Y)
80 even_Y = (all_X - all_Y)
81 The sets of points with equal hypotenuse are the same in the even and
83 all, but the flip takes them in reverse order. The first such reversal
84 occurs at N=14 and N=15. In even-points they're at 7,1 and 5,5. In
85 all-points they're at 5,0 and 4,3 and those two map 5,5 and 7,1, ie.
86 the opposite way around.
87 Odd Points
89 Option "points => "odd"" visits just the odd points, meaning sum X+Y
90 odd, so X,Y one odd the other even.
91 12 | 66
93 11 | points => "odd" 57
94 10 | 47 58
95 9 | 39 49
96 8 | 32 41 51
97 7 | 25 33 42
98 6 | 20 26 35 45
99 5 | 14 21 29 37
100 4 | 10 16 22 30 40
101 3 | 7 11 17 24 34
102 2 | 4 8 13 19 28 38
103 1 | 2 5 9 15 23 31
104 Y=0 | 1 3 6 12 18 27 36
105 +------------------------------------------
106 X=0 1 2 3 4 5 6 7 8 9 10 11 12 13
107 The X=Y diagonal is excluded because it has X+Y even.
109
111 See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
112 classes.
113 "$path = Math::PlanePath::HypotOctant->new ()"
115 "$path = Math::PlanePath::HypotOctant->new (points => $str)"
116 Create and return a new hypot octant path object. The "points"
117 option can be
118 "all" all integer X,Y (the default)
120 "even" only points with X+Y even
121 "odd" only points with X+Y odd
122 "($x,$y) = $path->n_to_xy ($n)"
124 Return the X,Y coordinates of point number $n on the path.
125 For "$n < 1" the return is an empty list, it being considered the
127 first point at X=0,Y=0 is N=1.
128 Currently it's unspecified what happens if $n is not an integer.
130 Successive points are a fair way apart, so it may not make much
131 sense to give an X,Y position in between the integer $n.
132 "$n = $path->xy_to_n ($x,$y)"
134 Return an integer point number for coordinates "$x,$y". Each
135 integer N is considered the centre of a unit square and an "$x,$y"
136 within that square returns N.
137
139 The calculations are not very efficient currently. For each Y row a
140 current X and the corresponding hypotenuse X^2+Y^2 are maintained. To
141 find the next furthest a search through those hypotenuses is made
142 seeking the smallest, including equal smallest, which then become the
143 next N points.
144 For "n_to_xy()" an array is built in the object used for repeat
146 calculations. For "xy_to_n()" an array of hypot to N gives a the first
147 N of given X^2+Y^2 distance. A search is then made through the next
148 few N for the case there's more than one X,Y of that hypot.
149
151 Entries in Sloane's Online Encyclopedia of Integer Sequences related to
152 this path include
153 <http://oeis.org/A024507> (etc)
155 points="all"
157 A024507 X^2+Y^2 of all points not on X axis or X=Y diagonal
158 A024509 X^2+Y^2 of all points not on X axis
159 being integers occurring as sum of two non-zero squares,
160 with repetitions for multiple ways
161 points="even"
163 A036702 N on X=Y leading Diagonal
164 being count of points norm<=k
165 points="odd"
167 A057653 X^2+Y^2 values occurring
168 ie. odd numbers which are sum of two squares,
169 without repetitions
170
172 Math::PlanePath, Math::PlanePath::Hypot,
173 Math::PlanePath::TriangularHypot, Math::PlanePath::PixelRings,
174 Math::PlanePath::PythagoreanTree
175
177 <http://user42.tuxfamily.org/math-planepath/index.html>
178
180 Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020
181 Kevin Ryde
182 This file is part of Math-PlanePath.
184 Math-PlanePath is free software; you can redistribute it and/or modify
186 it under the terms of the GNU General Public License as published by
187 the Free Software Foundation; either version 3, or (at your option) any
188 later version.
189 Math-PlanePath is distributed in the hope that it will be useful, but
191 WITHOUT ANY WARRANTY; without even the implied warranty of
192 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
193 General Public License for more details.
194 You should have received a copy of the GNU General Public License along
196 with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
197perl v5.36.0 2022-07-22 Math::PlanePath::HypotOctant(3)