NAME

Math::PlanePath::HypotOctant -- octant of points in order of hypotenuse distance

SYNOPSIS

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

DESCRIPTION

This path visits an octant of integer points X,Y in order of their distance from the origin 0,0. The points are a rising triangle 0<=Y<=X,

     8                                   61
     7                               47  54
     6                           36  43  49
     5                       27  31  38  44
     4                   18  23  28  34  39
     3               12  15  19  24  30  37
     2            6   9  13  17  22  29  35
     1        3   5   8  11  16  21  26  33
    Y=0   1   2   4   7  10  14  20  25  32  ...

         X=0  1   2   3   4   5   6   7   8
  

For example N=11 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).

In general the X,Y points are the sums of two squares X^2+Y^2 taken in increasing order of that hypotenuse, but only the "primitive" X,Y combinations, primitive in the sense of excluding mere negative X or Y or swapped Y,X.

Equal Distances

Points with the same distance from the origin are taken in anti-clockwise order from the X axis, which means by increasing Y. Points the same distance arise when there's more than one way to express a given distance as the sum of two squares.

Pythagorean triples give a point on the X axis and also above it at the same distance. For example 5^2 == 4^2 + 3^2 has N=14 at X=5,Y=0 and N=15 at X=4,Y=3, both 5 away from the origin.

Combinations like 20^2 + 15^2 == 24^2 + 7^2 occur too, and also with three or more different ways to have the same sum distance.

FUNCTIONS

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

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

Create and return a new hypot octant 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, 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.

FORMULAS

The calculations are not very efficient currently. For each Y row a current X and the corresponding hypotenuse X^2+Y^2 are maintained. To find the next furthest a search through those hypotenuses is made seeking the smallest, including equal smallest, which then become the next N points.

For n_to_xy an array is built and re-used for repeat calculations. For xy_to_n an array of hypot to N gives a the first N of given X^2+Y^2 distance. A search is then made through the next few N for the case there's more than one X,Y of that hypot.

SEE ALSO

Math::PlanePath, Math::PlanePath::Hypot, Math::PlanePath::PixelRings, Math::PlanePath::PythagoreanTree

HOME PAGE

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

LICENSE

Copyright 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/>.