Quartic equation of a spindle hyperboloid

2c = 2ae, the distance between the foci of the hyperbola, must equal the field radius plus the oculus radius.

When the parameters a, b, c and e are given their usual meanings, the equation of a hyperbola is:

x*x/a*a - y*y/b*b = 1

with

c = a*e

e = c/a

and

a*a + b*b = c*c .

(At a given c, the closer the vertex lies to the center, the more eccentric the hyperbola. An infinitely eccentric hyperbola is a straight line through the center.)

In designing a spindle hyperboloid lens for Veselago beam-down optics, the parameter that must be fixed at the outset is the distance, D, that separates the two foci,

D = 2c = heliostat field radius + oculus radius.

Once we fix c at c = D/2, variation in the eccentricity of the hyperbola varies the radius of the lamp.

The offset, R, of the rotational axis from the center of the hyperbola is

R = c - oculus radius .

Given an offset R, the quartic equation for a spindle hyperboloid can be derived as follows:

Derivation of the quartic equation for a spindle hyperboloid.

In a POV-Ray scene description a spindle hyperboloid Veselago lens becomes something like:

#declare a = 1.0; //distance from hyperbola center to vertex
#declare b = 1.0; //asymptote of hyperbola slope = b/a
#declare a2 = a*a;
#declare b2= b*b;
#declare ab2 = a2/b2;
#declare R = 1.8; // offset of hyperboloid rotational axis
#declare R2 = R*R;
….
 quartic {
//     x^4   x^3y   x^3z   x^3   x^2y^2
     < 1,    0,     0,     0,    -2*ab2,
//     x^2yz  x^2y   x^2z^2   x^2z   x^2
       0,    0,    2,     0,   2*R2-2*a2-4*R2,
//     xy^3  xy^2z  xy^2   xyz^2  xyz
       0,    0,     0,     0,    0,
//     xy    xz^3   xz^2   xz    x
       0,    0,     0,     0,    0,
//     y^4   y^3z   y^3    y^2z^2  y^2z
      ab2*ab2,    0,     0,    -2*ab2,   0,
//     y^2   yz^3   yz^2   yz    y
       2*a2*ab2-2*R2*ab2,    0,    0,    0,   0
//     z^4   z^3    z^2    z     Const
       1,    0,     2*R2-2*a2-4*R2,   0,   R2*R2-2*R2*a2+a2*a2 >
           clipped_by
    {sphere {<0,0,0>1.0 }}
   bounded_by{clipped_by}
    sturm
    pigment {Clear}
  interior { ior -1.0 }
  scale <100.0,100.0, 100.0>
  translate<0,0,0>
  }