Given three lines $l_a, l_b, l_c$ in $\mathbb {R}^3$ and three positive numbers $a, b, c>0$ I would like to find points $A, B, C$ on $l_a, l_b, l_c$ respectively, such that the side lengths of triangle $ABC$ are $a, b, c$. I know that this problem can not always have a solution and the solution is not necessarily unique but in my case I know that there is a solution and I know that I can start with a good initial guess. One possibility is to do an iteration, however I wonder if there is an analytical solution.
At the end my goal is to estimate the pose of three lines from three intersections with a sphere. This problem can be reduced to the problem above.
The 2-dimensional case would also be interesting. Furthermore I am also interested in the case where the three lines $l_a, l_b, l_c$ intersect in one point.