Sphere Packing inside a quadratic space

Question

How would you go about calculating the number of uniform spheres that could fit inside a given quadratic surface , say an elliptic paraboloid of a fixed height, given the radius of said spheres and the equation of the quadratic surface? Assuming two scenarios where the spheres are loosely packed and again when the spheres have the maximum kissing number.

Answer 1

Since you are asking for a maximal packing the question is likely very hard since the question for how to maximally pack monodisperse spheres in $\mathbb{R}^3$ was open until Hales solved it in 1998:

https://en.wikipedia.org/wiki/Kepler_conjecture