Minimize the distance in the Euclidean space

Question

The objective is to minimise the distance $d_{0}+d_{1}$. The points $c_{0}$ and $c_{1}$ are given. I need to locate the point $c$ which minimises the distance $d_{0}+d_{1}$.

I have worked like this. To minimise the distance, The point $c$ which is in the horizontal plane makes same angle with the points $c_{0}$ and $c_{1}$ ($\theta$ in my picture) and $c$ can be moved along the $z$ axis direction. The problem is how to find the coordiantes of $c$ which means $x,y,z$?

Answer 1

Be careful, the two angles may not be the same! I would find an expression for the two distances just using coordinates and substitute in any relationships you know arising from the point being on the plane, etc. and then use calculus to minimise the total distance.

Answer 2

I think this is well known. Reflect $c_1 $ and the line connecting $c$ and $c_1$ in the given plane, and apply the triangle inequality. The line connecting $c_0$ and the reflected $c_1$ intersects the plane in the optimum $c$.