--- name: "Math::PlanePath::Hypot" section: "3pm" platform: "linux" package: "libmath-planepath-perl" version: "129-1" release: "Debian 13 (Trixie)" platforms: ["linux"] url: "https://new.carta.tech/Math::PlanePath::Hypot" --- # Math::PlanePath::Hypot(3pm) ## 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 \ (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 \ ## 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 \.