Re: Douady-Hubbard Potential

Sorry, as usual some typos:

1] "For a perfect circle-shaped wire, the potential is radial and equal
to log|z|."

but should be

"For a perfect circle-shaped wire, the potential is radial and
equal to log|c|."

2] property . G(Zn+1) = G(Z)

should be

. G(Zn+1) = 2G(Z)


The code for the potential pictures could be something like (all
variables are floats)

// c = a + ib

x = x2 = 0.0f;
y = y2 = 0.0f;

sum = 0.00f;
fac = 0.25f;
for( int i=0; i<512; i++ )
{
// z^2 + c
x += a;
y += b;

x2 = x*x;
y2 = y*y;

// z -> z^2
y = 2.0f*x*y;
x = x2-y2;

// 1+ c/z^2
d = 1.0f/(x*x + y*y);
tx = ( a*x + b*y)*d + 1.0f;
ty = (-a*y + b*x)*d;

// increment summation (note log(sqrt(x)) = .5*log(x)
sum += fac*logf( tx*tx+ty*ty );
fac *= .5f;

// test escape radious 10
m2 = x2+y2;
if( m2>100.0f )
break;
}

G = .5f*logf(a*a+b*b) + sum;

But this is ugly (an attempt to optimize). If you have a complex
number library, this would be the "clean" version

z = 0;
G = log(abs(c));
fac = 0.5;
for( i=0; i<512; i++ )
{
// iterate
z = z*z + c;
// accumulate potential
G += fac*log(abs(1+c/(z*z));
// test escape radious 10
if( abs(z)>10 )
break;
}

.