How to find curve which runs through complex points?

Question

I have lists of complex points: orbit of complex point z under quadratic function

 f(z) = z*z

I know that lists are:

• z, z^2, z^4, z^8, ...
• (r,t), (r^2, 2*t), ..., (r^(2^n), t*2^n)

where :

• r = abs(z)
• t = arg(z)

So I think that these curve will be exponential spirals.

But my code:

 GiveParametric(radius,tMin,tMax) := 
 parametric(radius^t*cos(t),radius^t*sin(t),t,tMin,tMax)$

 GivePolar(radius, tMin,tMax) :=  polar(radius^(2^t),t,tMin,tMax)$

does not work.

Here is the image of 3 orbits (lists). Each list sould have it's own curve ( function)

Question :

How to draw (or find equations of) curves which runs through these points ?

Answer 1

I have used definition to draw sequence of points joined by lines

GiveContOrbit(r0,a0,tMin, tMax, dt ):=
 block(
   [Orbit,a,r,t, b],
   t : tMin,
   b: 2^t,
   a:a0*b,
   r: r0^b,
   z: GiveZ(r,a),
   Orbit:[[realpart(z),imagpart(z)]], 
   for t:tMin thru tMax step dt do
    ( 
        b: 2^t,
    a:a0*b,
    r: r0^b,
    z: GiveZ(r,a), 
    Orbit:endcons([realpart(z),imagpart(z)],Orbit)),
    return(Orbit) 
    )$

This in not what I wanted but seems to be a good aproximation. As I see curves intersects.