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 anticlockwise 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 anticlockwise 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 allpoints. And conversely N=1,2,10,26,etc on the X=Y diagonal here is the X axis of allpoints.
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 X1,Y+1 to put N=1 at the origin. The effect of that shift is as if the hypot measure in ``all'' points was (X1/2)^2+(Y1/2)^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
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016 Kevin RydeThis file is part of MathPlanePath.
MathPlanePath 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.
MathPlanePath 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 MathPlanePath. If not, see <http://www.gnu.org/licenses/>.