Provided by: libmath-planepath-perl_129-1_all 
NAME
Math::PlanePath::Hypot -- points in order of hypotenuse distance
SYNOPSIS
use Math::PlanePath::Hypot;
my $path = Math::PlanePath::Hypot->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This path visits integer points X,Y in order of their distance from the origin 0,0, or
anti-clockwise from the X axis among those of equal distance,
84 73 83 5
74 64 52 47 51 63 72 4
75 59 40 32 27 31 39 58 71 3
65 41 23 16 11 15 22 38 62 2
85 53 33 17 7 3 6 14 30 50 82 1
76 48 28 12 4 1 2 10 26 46 70 <- Y=0
86 54 34 18 8 5 9 21 37 57 89 -1
66 42 24 19 13 20 25 45 69 -2
77 60 43 35 29 36 44 61 81 -3
78 67 55 49 56 68 80 -4
87 79 88 -5
^
-5 -4 -3 -2 -1 X=0 1 2 3 4 5
For example N=58 is 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).
See "TriangularHypot" for points in order of X^2+3*Y^2, or "DiamondSpiral" and
"PyrmaidSides" in order of plain sum X+Y.
Equal Distances
Points with the same distance are taken in anti-clockwise order around from the X axis.
For example X=3,Y=1 is sqrt(10) from the origin, as are the swapped X=1,Y=3, and X=-1,Y=3
etc in other quadrants, for a total 8 points N=30 to N=37 all the same distance.
When one of X or Y is 0 there's no negative, so just four negations like N=10 to 13 points
X=2,Y=0 through X=0,Y=-2. Or on the diagonal X==Y there's no swap, so just four like N=22
to N=25 points X=3,Y=3 through X=3,Y=-3.
There can be more than one way for the same distance to arise. A Pythagorean triple like
3^2 + 4^2 == 5^2 has 8 points from the 3,4, then 4 points from the 5,0 giving a total 12
points N=70 to N=81. Other combinations like 20^2 + 15^2 == 24^2 + 7^2 occur too, and
also with more than two different ways to have the same sum.
Multiples of 4
The first point of a given distance from the origin is either on the X axis or somewhere
in the first octant. The row Y=1 just above the axis is the first of its equals from X>=2
onwards, and similarly further rows for big enough X.
There's always a multiple of 4 many points with the same distance so the first point has
N=4*k+2, and similarly on the negative X side N=4*j, for some k or j. If you plot the
prime numbers on the path then those even N's (composites) are gaps just above the
positive X axis, and on or just below the negative X axis.
Circle Lattice
Gauss's circle lattice problem asks how many integer X,Y points there are within a circle
of radius R.
The points on the X axis N=2,10,26,46, etc are the first for which X^2+Y^2==R^2 (integer
X==R). Adding option "n_start=>0" to make them each 1 less gives the number of points
strictly inside, ie. X^2+Y^2 < R^2.
The last point satisfying X^2+Y^2==R^2 is either in the octant below the X axis, or is on
the negative Y axis. Those N's are the number of points X^2+Y^2<=R^2, Sloane's A000328.
When that A000328 sequence is plotted on the path a straight line can be seen in the
fourth quadrant extending down just above the diagonal. It arises from multiples of the
Pythagorean 3^2 + 4^2, first X=4,Y=-3, then X=8,Y=-6, etc X=4*k,Y=-3*k. But sometimes the
multiple is not the last among those of that 5*k radius, so there's gaps in the line. For
example 20,-15 is not the last since because 24,-7 is also 25 away from the origin.
Even Points
Option "points => "even"" visits just the even points, meaning the sum X+Y even, so X,Y
both even or both odd.
points => "even"
52 40 39 51 5
47 32 23 31 46 4
53 27 16 15 26 50 3
33 11 7 10 30 2
41 17 3 2 14 38 1
24 8 1 6 22 <- Y=0
42 18 4 5 21 45 -1
34 12 9 13 37 -2
54 28 19 20 29 57 -3
48 35 25 36 49 -4
55 43 44 56 -5
^
-5 -4 -3 -2 -1 X=0 1 2 3 4 5
Even points can be mapped to all points by a 45 degree rotate and flip. N=1,6,22,etc on
the X axis here is on the X=Y diagonal of all-points. And conversely N=1,2,10,26,etc on
the X=Y diagonal here is the X axis of all-points.
The sets of points with equal hypotenuse are the same in the even and all, but the flip
takes them in a reversed order.
Odd Points
Option "points => "odd"" visits just the odd points, meaning sum X+Y odd, so X,Y one odd
the other even.
points => "odd"
71 55 54 70 6
63 47 36 46 62 5
64 37 27 26 35 61 4
72 38 19 14 18 34 69 3
48 20 7 6 17 45 2
56 28 8 2 5 25 53 1
39 15 3 + 1 13 33 <- Y=0
57 29 9 4 12 32 60 -1
49 21 10 11 24 52 -2
73 40 22 16 23 44 76 -3
65 41 30 31 43 68 -4
66 50 42 51 67 -5
74 58 59 75 -6
^
-6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6
Odd points can be mapped to all points by a 45 degree rotate and a shift X-1,Y+1 to put
N=1 at the origin. The effect of that shift is as if the hypot measure in "all" points
was (X-1/2)^2+(Y-1/2)^2 and for that reason the sets of points with equal hypots are not
the same in odd and all.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$path = Math::PlanePath::Hypot->new ()"
"$path = Math::PlanePath::Hypot->new (points => $str), n_start => $n"
Create and return a new hypot path object. The "points" option can be
"all" all integer X,Y (the default)
"even" only points with X+Y even
"odd" only points with X+Y odd
"($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 say 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,$y" within that square returns N.
For "even" and "odd" options only every second square in the plane has an N and if
"$x,$y" is a position not covered then the return is "undef".
FORMULAS
The calculations are not particularly efficient currently. Private arrays are built
similar to what's described for "HypotOctant", but with replication for negative and
swapped X,Y.
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
<http://oeis.org/A051132> (etc)
points="all", n_start=0
A051132 N on X axis, being count points norm < X^2
points="odd"
A005883 count of points with norm==4*n+1
SEE ALSO
Math::PlanePath, Math::PlanePath::HypotOctant, 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/>.