# Is there a name for the minimal surface connecting two straight line segments in 3-dim Euclidean space?

In the $$2$$-dim Euclidean plane, the minimal curve that joins (and necessarily contains) two given points is a straight line.

Here what is being minimized by the curve is the $$1$$-dim measure of the $$1$$-dim object in the $$2$$-dim space that joins the two given $$0$$-dim objects.

In $$3$$-dim Euclidean space, what is the minimal surface that connects (and necessarily contains) two given straight line segments (of known finite length)?

Again, here what is being minimized is the surface area ($$2$$-dim measure) of the $$2$$-dim object (embedded in $$3$$-dim space) connecting two given $$1$$-dim objects.

Is there an umbrella name for this whole category?

By "the whole category" I mean given line segments can be in general configurations and not just special cases like being parallel, or the two segments having the same length. (btw the special case of parallel and same length seems to yield helicoid as the minimal surface)

As pointed by Rahul in the comment, besides the given two straight line segments, it is necessary that one specifies the "other" boundaries. For simplicity, a reasonable choice seems to be connecting the end points to have a non-planar (skew) quadrilateral frame (for the soap film, so to speak).

Note: Just found this, which doesn't solve my problem (give me the keyword I'm looking for) but might contain information to references that does. Looking into it right now.

I suspect this should be one of the basic stuff in differential geometry that everybody knows or agrees on (just like how people agree on the notion of a straight line). The next section describes my current guesswork, and it would be nice to know if I'm on the right track.

## Guess: the unique ruled surface

Given two straight line segments in the $$3$$-dim space, it suffice to consider the two skew lines $$L_1$$ and $$L_2$$ that contain them respectively. As one can parametrize a segment as part of an infinite line, the desired minimal surface from the given segments is part of the surface "generated" by $$L_1$$ and $$L_2$$.

One can find the unique pair of points, $$P_1$$ on $$L_1$$ and $$P_2$$ on $$L_2$$, that have the smallest distance. Accordingly one obtains $$\overset{\longrightarrow}{P_1P_2}$$ as the common normal vector between for two skew lines.

Now one can define a ruled surface by parametrizing the "moving" lines $$L(t)$$, thus making a ruled surface, with parameter $$t$$: starting with $$L(t) = L_1$$ at $$t = 0$$, "move" $$L(t)$$ towards $$L_2$$ along $$\overset{\longrightarrow}{P_1P_2}$$, "anchoring" at $$P_1$$ and rotate $$L(t)$$ along the way to make it eventually coincide with $$L_2$$.

The surface is not yet unique before specifying the amount of rotation (with respect to the distance to $$L_2$$). I'm guessing the rotation should be done linearly, but I haven't figure out if this indeed minimizes the surface area.

Obviously a standard approach is to formulate the whole thing in terms of calculus of variations.

I've done the almost trivial case of a straight line minimizing the distance, but apparently things get very complicated already for a $$2$$-dim surface in $$3$$-dim.

Even in the special cases where the coordinates are setup nicely so that the surface is a function like $$z = f(x,y)$$, it's an elliptic PDE and the solution seems to vary greatly depending on the boundary conditions. In short, I don't know how to handle the situation.

• I think you need to connect the two pairs of endpoints to get a quadrilateral. Otherwise the surface can reduce its area merely by shrinking its boundary curves inwards. – Rahul Nov 15 '18 at 4:06
• Thanks for the input. I shall do a quick edit to the post just for now, as clearly I have missed this very important point so far that might invalidate most of the current content. – Lee David Chung Lin Nov 15 '18 at 4:17