4
$\begingroup$

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.

$\endgroup$
1
  • 1
    $\begingroup$ Suppose the lines are (1+2r,0,3r), (-1+4s,5s,0) and (0,1,t), and the side lengths are 6,7,8. Then it’s easy to write out the equations and eliminate t, and the result convinces me that there’s no analytical solution: the equation for r would be at least of 8th degree. $\endgroup$
    – user44143
    Nov 3, 2017 at 15:30

1 Answer 1

2
$\begingroup$

This is just confirming and illustrating @MattF.'s example of three skew lines in $\mathbb{R}^3$. Solving the equations yields $8$ $(r,s,t)$ solutions for a $(6,7,8)$ triangle: $4$ imaginary, and $4$ real.


          TriOnLines
Black point: origin. Three lines: red, blue, green, $(1+2r,0,3r), (-1+4s,5s,0), (0,1,t)$. Four solution $\triangle$s.
Although, as he says, there is (in general) no analytical solution, numerical solutions are easily computed.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.