Math::PlanePath::VogelFloret -- circular spiral like a sunflower
use Math::PlanePath::VogelFloret; my $path = Math::PlanePath::VogelFloret->new; my ($x, $y) = $path->n_to_xy (123);
The Vogel spiral arranges integer points in a spiraling pattern so they align to resemble the pattern of seeds found in the head of a sunflower.
The polar coordinates are
R = sqrt(N) * FACTOR theta = (N / (PHI**2)) * 2pi
where PHI is the golden ratio (1+sqrt(5))/2 and FACTOR is a scaling factor of about 1.6 designed to put the points 1 apart (or a little more).
Most of the other PlanePaths are implicitly quadratic, but the VogelFloret is instead essentially based on near-integer multiples of PHI**2 (which is PHI+1)..
The fibonacci numbers fall close to the X axis to the right because they're roughly powers of the golden ratio, F(k) ~= (PHI**k)/sqrt(5). The exponential grows faster than the sqrt in the R radial distance so they soon become widely spaced though. The Lucas numbers similarly.
$path = Math::PlanePath::VogelFloret->new ()
Create and return a new path object.
($x,$y) = $path->n_to_xy ($n)
Return the x,y coordinates of point number
$non the path.
$ncan be any value
$n >= 0and fractions give positions on the spiral in between the integer points.
$n < 0the return is an empty list, it being considered there are no negative points in the spiral.
$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 circle of diameter 1 and an
$x,$ywithin that circle returns N.
The path is scaled so no two points are closer than 1 apart so the circles don't overlap, but they also don't cover the plane and if
$x,$yis not within one of those circles then the return is
Math-PlanePath is Copyright 2010 Kevin Ryde
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.
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