Closest point of line segment without endpoints

I know of a formula to determine shortest line segment between two given line segments, but that works only when endpoints are included. I'd like to know if there is a solution when endpoints are not included or if I'm mixing disciplines incorrectly.

Example : Line segment $A$ is from $(1, 1)$ to $(1, 4)$ and line segment $B$ is from $(0, 0)$ to $(0, 2)$, so shortest segment between them would be $(0, 1)$ to $(1, 1)$. But of line segment $A$ did not include those end points, how would that work since $(1, 1)$ is not part of line segment $A$?

• I thank the quick comments from Millikan, Nicolas, and Smith. This is for a software library, and I'm finding the line segments without end points to be not working well. – MJ5 Jun 25 '12 at 20:55

There would not be a shortest line segment. Look at line segments from $(0,1)$ to points very close to $(1,1)$ on the segment that joins $(1,1)$ and $(1,4)$. These segments get shorter and shorter, approaching length $1$ but never reaching it.
In your example, if the endpoints are not included in $A$ and $B$, then there is no shortest line segment connecting the two, since for any supposedly shortest path someone gives you, you can just slide the top end a teeny bit more downward in the direction of $(1,1)$.
In this case. there is no shortest segment. If you make the segment from $(0,1)$ to $(1,1+ \epsilon)$, a shorter one is from $(0,1)$ to $(1,1+ \epsilon /2)$