Given that all 3 lines are infinitely long, how do I find the smallest sphere that intersects (or touches) those 3 lines in 3D?
There is a simple case where the sphere can be calculated from just 2 lines and the third line can just pass through the sphere. You can ignore that case.
The case I want to solve here is when the distances from the sphere center to all 3 lines are equal (all the lines are tangent to the sphere surface). Trying to solve this case leads me to the path of solving quadratic equation systems and Lagrange multiplier which I cannot solve for an exact closed-form solution.
As I want this solution to be efficient, I try to avoid iterative methods/numerical optimizations.