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Math::PlanePath::HypotOctant -- octant of points in order of hypotenuse distance
Contents
Description
This path visits an octant of integer points X,Y in order of their distance from the origin 0,0. The
points are a rising triangle 0<=Y<=X,
8 | 61
7 | 47 54
6 | 36 43 49
5 | 27 31 38 44
4 | 18 23 28 34 39
3 | 12 15 19 24 30 37
2 | 6 9 13 17 22 29 35
1 | 3 5 8 11 16 21 26 33
Y=0 | 1 2 4 7 10 14 20 25 32 ...
+---------------------------------------
X=0 1 2 3 4 5 6 7 8
For example N=11 at X=4,Y=1 is sqrt(4*4+1*1) = sqrt(17) from the origin. The next furthest from the
origin is X=3,Y=3 at sqrt(18).
This octant is "primitive" elements X^2+Y^2 in the sense that it excludes negative X or Y or swapped Y,X.
EqualDistances
Points with the same distance from the origin are taken in anti-clockwise order from the X axis, which
means by increasing Y. Points with the same distance occur when there's more than one way to express a
given distance as the sum of two squares.
Pythagorean triples give a point on the X axis and also above. For example 5^2 == 4^2 + 3^2 has N=14 at
X=5,Y=0 simply as 5^2 = 5^2 + 0 and then N=15 at X=4,Y=3 for the triple. Both are 5 away from the
origin.
Combinations like 20^2 + 15^2 == 24^2 + 7^2 occur too, and also with three or more different ways to have
the same sum distance.
EvenPoints
Option "points => "even"" visits just the even points, meaning the sum X+Y even, so X,Y both even or both
odd.
12 | 66
11 | points => "even" 57
10 | 49 58
9 | 40 50
8 | 32 41 51
7 | 25 34 43
6 | 20 27 35 45
5 | 15 21 29 37
4 | 10 16 22 30 39
3 | 7 11 17 24 33
2 | 4 8 13 19 28 38
1 | 2 5 9 14 23 31
Y=0 | 1 3 6 12 18 26 36
+---------------------------------------
X=0 1 2 3 4 5 6 7 8 9 10 11 12
Even points can be mapped to all points by a 45 degree rotate and flip. N=1,3,6,12,etc on the X axis
here is on the X=Y diagonal of all-points. And conversely N=1,2,4,7,10,etc on the X=Y diagonal here is
on the X axis of all-points.
all_X = (even_X + even_Y) / 2
all_Y = (even_X - even_Y) / 2
even_X = (all_X + all_Y)
even_Y = (all_X - all_Y)
The sets of points with equal hypotenuse are the same in the even and all, but the flip takes them in
reverse order. The first such reversal occurs at N=14 and N=15. In even-points they're at 7,1 and 5,5.
In all-points they're at 5,0 and 4,3 and those two map 5,5 and 7,1, ie. the opposite way around.
OddPoints
Option "points => "odd"" visits just the odd points, meaning sum X+Y odd, so X,Y one odd the other even.
12 | 66
11 | points => "odd" 57
10 | 47 58
9 | 39 49
8 | 32 41 51
7 | 25 33 42
6 | 20 26 35 45
5 | 14 21 29 37
4 | 10 16 22 30 40
3 | 7 11 17 24 34
2 | 4 8 13 19 28 38
1 | 2 5 9 15 23 31
Y=0 | 1 3 6 12 18 27 36
+------------------------------------------
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13
The X=Y diagonal is excluded because it has X+Y even.
Formulas
The calculations are not very efficient currently. For each Y row a current X and the corresponding
hypotenuse X^2+Y^2 are maintained. To find the next furthest a search through those hypotenuses is made
seeking the smallest, including equal smallest, which then become the next N points.
For "n_to_xy()" an array is built in the object used for repeat calculations. For "xy_to_n()" an array
of hypot to N gives a the first N of given X^2+Y^2 distance. A search is then made through the next few
N for the case there's more than one X,Y of that hypot.
Functions
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$path = Math::PlanePath::HypotOctant->new ()"
"$path = Math::PlanePath::HypotOctant->new (points => $str)"
Create and return a new hypot octant path object. The "points" option can be
"all" all integer X,Y (the default)
"even" only points with X+Y even
"odd" only points with X+Y odd
"($x,$y) = $path->n_to_xy ($n)"
Return the X,Y coordinates of point number $n on the path.
For "$n < 1" the return is an empty list, it being considered the first point at X=0,Y=0 is N=1.
Currently it's unspecified what happens if $n is not an integer. Successive points are a fair way
apart, so it may not make much sense to give an X,Y position in between the integer $n.
"$n = $path->xy_to_n ($x,$y)"
Return an integer point number for coordinates "$x,$y". Each integer N is considered the centre of a
unit square and an "$x,$y" within that square returns N.
Home Page
License
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde
This file is part of Math-PlanePath.
Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the GNU
General Public License as published by the Free Software Foundation; either version 3, or (at your
option) any later version.
Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even
the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
License for more details.
You should have received a copy of the GNU General Public License along with Math-PlanePath. If not, see
<http://www.gnu.org/licenses/>.
perl v5.32.0 2021-01-23 Math::PlanePath::HypotOctant(3pm)
Name
Math::PlanePath::HypotOctant -- octant of points in order of hypotenuse distance
Oeis
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
<http://oeis.org/A024507> (etc)
points="all"
A024507 X^2+Y^2 of all points not on X axis or X=Y diagonal
A024509 X^2+Y^2 of all points not on X axis
being integers occurring as sum of two non-zero squares,
with repetitions for multiple ways
points="even"
A036702 N on X=Y leading Diagonal
being count of points norm<=k
points="odd"
A057653 X^2+Y^2 values occurring
ie. odd numbers which are sum of two squares,
without repetitions
See Also
Math::PlanePath, Math::PlanePath::Hypot, Math::PlanePath::TriangularHypot, Math::PlanePath::PixelRings,
Math::PlanePath::PythagoreanTree
Synopsis
use Math::PlanePath::HypotOctant;
my $path = Math::PlanePath::HypotOctant->new;
my ($x, $y) = $path->n_to_xy (123);
PNG Icons