NAME
Math::PlanePath::VogelFloret  circular spiral like a sunflower
SYNOPSIS
use Math::PlanePath::VogelFloret;
my $path = Math::PlanePath::VogelFloret>new;
my ($x, $y) = $path>n_to_xy (123);
DESCRIPTION
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 nearinteger 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.
FUNCTIONS
$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
$n
on the path.$n
can be any value$n >= 0
and fractions give positions on the spiral in between the integer points.For
$n < 0
the 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$x,$y
within 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
$x,$y
is not within one of those circles then the return isundef
.
SEE ALSO
Math::PlanePath, Math::PlanePath::SacksSpiral
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
MathPlanePath is Copyright 2010 Kevin Ryde
MathPlanePath 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.
MathPlanePath 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 MathPlanePath. If not, see <http://www.gnu.org/licenses/>.