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 $A$?

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.
