complex numbers and decimal powers
 by Adrian Porter


Here are the origional equations:
Z(n) = Z(n-1) - F(Z(n-1)) / F'(Z(n-1))
Z(0) = x + yi
F(i) = i ^ p - 1
and again we use the power rule (a * b ^ c -> a * c * b ^ (c-1)):
F'(i) = p * i ^ (p-1)

first, we must calculate F(Z(n-1)), we know Z(n-1) aka i
to raise a complex number (i) to a decimal power (p), we must:
i ^ p = |i|^p * cis(atanb(real(i), imag(i)))

definitions: 
cis(x) = cos(x) + i*sin(x)

atanb(y, x) = {	0		y=0 and x>=0
		pi		y=0
		pi/2		x=0 and y>=0
		3pi/2		x=0
		atan(y/x)	x>=0 and y>=0
		2pi+atan(y/x)	x>=0
		pi+atan(y/x)	else

|x| = sqr(real(x)^2 + imag(x)^2)

a * b = (real(a) * real(b) - imag(a) * imag(b)) + i(imag(a) * imag(b) + real(a) * real(b))

(foil)


here is the c++ code to do the iterations:


Zx = double(i)/(screenw-1)*curw+ax;
Zy = double(j)/(screenh-1)*curh+ay;

for(n = 1;n<= maxiterations; n++) {
				
	ZN=pow(distance(0,0,Zx,Zy), power); 
			
	theta=atanb(Zy, Zx);
	Fx1 = ZN * cos(power * theta) - 1;
	Fy1 = ZN * sin(power * theta);

				
	ZN=pow(distance(0,0,Zx,Zy), power - 1);
	Fx2 = power * ZN * cos((power - 1) * theta);
	Fy2 = power * ZN * sin((power - 1) * theta);

				
	ZN=pow(distance(0,0,Fx2,Fy2), -1);
	theta=atanb(Fy2, Fx2);
	Fx3 = ZN * cos(-1 * theta);
	Fy3 = ZN * sin(-1 * theta);


	Fx4 = Fx1 * Fx3 - Fy1 * Fy3;
	Fy4 = Fy1 * Fx3 + Fx1 * Fy3;

	rad=distance(Zx, Zy, Zx - Fx4, Zy - Fy4);

	if(rad <= .01) break;
	Zx -= Fx4; Zy -= Fy4;
}
for more information on that stuff: search altavista for "Eric's Treasure Trove"
back