/ 
Math-PlanePath-78
River stage one • 3 direct dependents • 4 total dependents
/ Math::PlanePath::TriangularHypot

NAME

Math::PlanePath::TriangularHypot -- points of triangular lattice in order of hypotenuse distance

SYNOPSIS

use Math::PlanePath::TriangularHypot;
my $path = Math::PlanePath::TriangularHypot->new;
my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

This path visits X,Y points on a triangular "A2" lattice in order of their distance from the origin 0,0 and anti-clockwise around from the X axis among those of equal distance.

         58    47    39    46    57                 4

      48    34    23    22    33    45              3

   40    24    16     9    15    21    38           2

49    25    10     4     3     8    20    44        1

   35    17     5     1     2    14    32      <- Y=0

50    26    11     6     7    13    31    55       -1

   41    27    18    12    19    30    43          -2

      51    36    28    29    37    54             -3

         60    52    42    53    61                -4

                      ^
-7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7

The lattice is put on a square X,Y grid using every second point per "Triangular Lattice" in Math::PlanePath. With scaling X/2, Y*sqrt(3)/2 to give equilateral triangles of side length 1 the distance from X,Y to the origin is

dist^2 = (X/2^2 + (Y*sqrt(3)/2)^2
       = (X^2 + 3*Y^2) / 4

For example N=19 at X=2,Y=-2 is sqrt((2**2+3*-2**2)/4) = sqrt(4) from the origin. The next smallest after that is X=5,Y=1 at sqrt(7). The key part is X^2 + 3*Y^2 as the distance measure to order the points.

Equal Distances

Points with the same distance are taken in anti-clockwise order around from the X axis. For example N=14 at X=4,Y=0 is sqrt(4) from the origin, as are the rotated X=2,Y=2 and X=-2,Y=2 etc in other sixths, for a total 6 points N=14 to N=19 all the same distance.

In general there's either 6 or 12 symmetric points so the number of points of the same distance is always a multiple of 6 or 12. There are 6 points when on the six radial lines X=0, X=Y or X=-Y, or on the lines Y=0, X=3*Y or X=-3*Y which are midway between them. Then there's 12-way symmetry for anything else, ie. anything in the twelve slices between those twelve lines. For example the first 12 equal is N=20 to N=31 all at sqrt(28).

There can also be further ways for the same distance to arise, but the 6-way or 12-way symmetry means always a multiple of 6 or 12.

FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

$path = Math::PlanePath::TriangularHypot->new ()

Create and return a new triangular hypot path object.

($x,$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 as 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.

Only every second square in the plane has an N, being those where X,Y both odd or both even. If $x,$y is a position without an N, ie. one of X,Y odd the other even, then the return is undef.

OEIS

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include,

http://oeis.org/A035019

A003136  norms X^2+3*Y^2 which occur
A004016  count of points of norm n
A035019    skipping zero counts
A088534    counting only in the twelfth 0<=X<=Y

The counts in these sequences are expressed as norm = x^2+x*y+y^2. That x,y is related to the "even" X,Y on the path here by a -45 degree rotation,

x = (Y-X)/2           X = 2*(x+y)
y = (X+Y)/2           Y = 2*(y-x)

The norm is then

norm = x^2+x*y+y^2
     = ((Y-X)/2)^2 + (Y-X)/2 * (X+Y)/2 + ((X+Y)/2)^2
     = (X^2 + 3*Y^2) / 4

X^2+3*Y^2 is the dist^2 described above for equilateral triangles of unit side. The factor of /4 doesn't affect the count of how many points.

Sequences A092572, A092573 and A158937 are based on x^2+3*y^2 but they're not applicable to this TriangularHypot since they're all integer x,y whereas the path here is every second point, ie. x,y both odd or both even. The latter condition gives the x^2+x*y+y^2 form.

SEE ALSO

Math::PlanePath, Math::PlanePath::Hypot, Math::PlanePath::HypotOctant, Math::PlanePath::PixelRings, Math::PlanePath::HexSpiral

HOME PAGE

http://user42.tuxfamily.org/math-planepath/index.html

LICENSE

Copyright 2010, 2011, 2012 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/>.