So, here's the deal - the equations you're suggesting are AFAIK, going to be
_parametric_.   x, y, and z are going to depend on some independent variable, in
your case, t.

So the easiest way to implement this is with a parametric {} object.
It's also ridiculously slow unless you employ some tricks.

Parametrics use the variables u and v.   I used u for the curve, and v for the
height.

An isosurface would require you to figure out what the implicit equation is for
the curve - one equation that uses x, y, and z.

Not sure what your idea is with this - I could speculate, but I'd say there are
probably better ways.



#version 3.7;
// Bill Walker "Bald Eagle" March 2018

global_settings {assumed_gamma 1.0}
#include "colors.inc"

camera {
                            location  <0, 10, -20>
                            right    x*image_width/image_height
                            look_at   <0, 0, 0>}

light_source { <0, 15, -50>  color rgb <1, 1, 1>}

plane {y, 0 pigment {Gray10}}

#declare a = 0.5;  //(1+sqrt(5))/2;
#declare r = 0.25;

parametric {
 function {r * (cos (u) + (u - a) * sin (u))}
 function { v }
 function {r * (sin (u) - (u - a) * cos (u))}

 <0, -1>, <10*tau, 1>  // start, end of (u,v)

  contained_by {box {<1, 1, 1>*-10, <1, 1, 1>*10} }
  max_gradient 20
  accuracy 0.01
  precompute 20 x,y,z

  texture {pigment {color rgb <1,1,1>}
           finish {specular 0.4 phong 0.5}}
  scale 1
  rotate <0, 0, 0>
}

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