NAME
Math::PlanePath::HypotOctant -- octant of points in order of hypotenuse distance
SYNOPSIS
use Math::PlanePath::HypotOctant;
my $path = Math::PlanePath::HypotOctant->new;
my ($x, $y) = $path->n_to_xy (123);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 8For 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.
Equal Distances
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.
Even Points
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 12Even 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.
Odd Points
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 13The X=Y diagonal is excluded because it has X+Y even.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::HypotOctant->new ()$path = Math::PlanePath::HypotOctant->new (points => $str)-
Create and return a new hypot octant path object. The
pointsoption can be"all" all integer X,Y (the default) "even" only points with X+Y even "odd" only points with X+Y odd ($x,$y) = $path->n_to_xy ($n)-
Return the X,Y coordinates of point number
$non the path.For
$n < 1the 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
$nis not an integer. Successive points are a fair way apart, so it may not make much sense to give an X,Y position in between the integer$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$x,$ywithin that square returns N.
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.
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 repetitionsSEE ALSO
Math::PlanePath, Math::PlanePath::Hypot, Math::PlanePath::TriangularHypot, Math::PlanePath::PixelRings, Math::PlanePath::PythagoreanTree
HOME PAGE
http://user42.tuxfamily.org/math-planepath/index.html
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/>.