I How to model and resolve a static non-interpenetration constraint

[fermipasta]

*Constraints (sorry the title got mangled)

I want to model N spherical points p_i in R³ with masses m_i and bounding radii of r_i for 1 <= i <= N.

So I can write (N²-N)/2 inequality constraints:

C_i,j : ||p_i - p_j|| - r_i - r_j >= 0

for unique pairs of i,j.

Looking at the same problem for three independent "spheres" (intervals) in 1D, it is possible to visualize clearly since the configuration space is then R³, the solution is a geometric projection to the "planes" defined by each unique pair, and to take into account the mass is just to project the point in configuration space along the "mass plane" which is perpendicular to the "mass vector" (m₁, m₂, m₃) to the solution. The solution (if the configuration point is inside one of these forbidden regions) is the nearest "external" point defined by these planes intersecting in the configuration space. This is not too hard to do, it's either the nearest plane (one contact) or a line defined by the intersection of two planes (two contacts).

I'm not sure how think about the configuration space for N spheres in R³. A single pair of spheres would create a 6D configuration space. My thinking may be: if two spheres are in contact, they have either 4 or 5 degrees of freedom in which they "stay in contact", so the configuration manifold for that constraint would be either 4D or 5D. Either sphere may move in two dimensions tangent to the contact point (4 total degrees of freedom). They could both move perpendicular to the contact in the same direction, maybe giving a 5th degree of freedom?

Anyway it seems (?) like a well-defined problem, I'm just not familiar with the usual formulation.

 

[fermipasta]

I guess the solution surface is not a hyperplane! If I write the constraint explicitly in terms of x,y,z:
$$
(x_1-x_2)^2 + (y_1-y_2)^2 + (z_1-z_2)^2 - C = 0
$$
I think it defines some kind of hypercylinder. So at least it seems that its not a system of linear equations, maybe it's a quadratic programming problem.