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...
*Constraints (sorry the title got mangled)
I want to model N spherical points pi in R3 with masses mi and bounding radii of ri for 1 <= i <= N.
So I can write (N2-N)/2 inequality constraints:
Ci,j : ||pi - pj|| - ri - rj >= 0
for unique pairs of i,j.
Looking at the same problem for three...