Math::PlanePath::HypotOctant -- octant of points in order of hypotenuse distance
use Math::PlanePath::HypotOctant; my $path = Math::PlanePath::HypotOctant->new; my ($x, $y) = $path->n_to_xy (123);
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.
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.
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.
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.
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.
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.
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
Math::PlanePath, Math::PlanePath::Hypot, Math::PlanePath::TriangularHypot, Math::PlanePath::PixelRings, Math::PlanePath::PythagoreanTree
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012, 2013, 2014 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.
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