Calculating normal of other quadratic objects

Hi,

First of all, my congratulations to the author of the book, topics are developed clearly (impossible to find the same quality in ray-tracing guides available on the internet).

After implementing, sphere, cone and cylinder, I had fun calculating the coefficients of other four quadratic figures:

• One sheet hyperboloid  _{f=x^2-y^2+z^2 - 1}
  
• Two sheets hyperboloid _{f=-x^2+y^2-z^2 - 1}
  
• Paraboloid elliptic _f=-x^2+z^2-y
  
• Paraboloid hyperbolic _f=-x^2-z^2-y

So, in object coordinate, it is clear to me that the normal of a sphere is 'the intersection point' and that the normal of a cylinder is 'the intersection point without the Y'.
The cylinder normal vector is less intuitive because I don't understand why Y is equal to sqrt(x * x + z * z) since all formulas that I can find on the internet for the cones aligned on the y axis are different.
Many articles suggest to take the gradient of F(x,y,z) to calculate the equation of the normal but for _f=x2+y2−z2, I obtain _{(2x,2y,−2z)} which is different of _{(x, sqrt(x*x+y*y), z)}.

Unfortunately, I find the same difficulty to calculate the normal vector with the new figures I  implemented.
Here are the coefficient formulas of hyperboloids if you are interested. (r = ray)
hyperboloid_1Sheet
Thank you,
Peppone