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 a scaling X/2, Y*sqrt(3)/2 to give equilateral triangles with side length 1 the X,Y distance from 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. 6 when on the six radial lines X=0, X=Y or X=-Y. And 6 also when on the six radial lines Y=0, X=3*Y or X=-3*Y (these are midway between the first six). And then 12 for anything in the twelve slices in between those lines, for example the first being N=20 through N=31 all at sqrt(28).

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

FUNCTIONS

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

Create and return a new 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.

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 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/>.