#! /usr/bin/perl
#
# This little perl script computes the locations of the 
# buds along the perimiter of the mandelbrot cardiod.
# 

print  " \n";
print  " guesstimate for the bud locations on the main sequence\n";
print  " accurate for the early part of the sequence, but fails long run\n";
print  " \n";
print  "n	phi		x-bud		y-bud		bud\n";
print  "			center		center		diameter\n";
print  "----    ------		------		------		--------\n";
for ($i=1; $i<=25; $i++) 
{

	# $nume is the Farey numerator
	$nume = 1;
	# $nume  = 2;

	# $deno is the Farey denominator
	$deno = $i+1;
	# $deno = 2*$i+1;

	# $t is natural parameter; $p is angle phi
   	$t = $nume/$deno;
	$p = 2*3.14159265358979*$t;

	# x,y is on the cardiod
	$x = 0.5 * cos ($p) - 0.25 * cos (2*$p);
	$y = 0.5 * sin ($p) - 0.25 * sin (2*$p);
	$r = sqrt ($x*$x+$y*$y);

	# $br is our guesstimate for the bud radius
	$xx = $x - 0.25;
	$br = sqrt ($xx*$xx+$y*$y);
	$br = sqrt ($br);
	$br *= 1 / ($deno*$deno);

	# note that $br == $brr is an exact identity.
 	$brr = sin ($p/2) / ($deno*$deno);

	# to get to the center of the bud, we need to compute 
	# the tangent to the cardiod.
	$dx = -0.5 * (sin ($p) - sin (2*$p));
	$dy = 0.5 * (cos ($p) - cos (2*$p));


	# $m is the slope of the line perpendicular to the cardiod
	# x,y should now point at the bud center.
	$m = - $dx/$dy;
	$l = sqrt (1+$m*$m);
	if ($x > 0) { $x += $br/$l; } else { $x -= $br/$l; }
	if ($y > 0) { $y += $m*$br/$l; } else { $y -= $m*$br/$l; }
	
	printf  "%3d	%8.6f	%8.6f	%8.6f	%8.6f\n", $i, $p, $x, $y, 2*$br;
}