there's my question:
Given 2 regular plane curves (let's say $\mathcal{C}^1$) in the 3D space, is there always a developable surface which contains both curves ?
Thanks, anders
there's my question:
Given 2 regular plane curves (let's say $\mathcal{C}^1$) in the 3D space, is there always a developable surface which contains both curves ?
Thanks, anders
I think the answer is no.
Take two skew lines, A and B as the two curves. Take some point P on line A. Using the "ruled surface" approach mentioned by J.M. this point P would be connected by a linear "ruling" to some point Q on line B. In order for the surface to be developable, the surface normal has to be constant along each such ruling. But clearly it won't be if the lines A and B are skew (not coplanar).
This isn't a rigorous proof. It only deals with attempts to construct a developable surface using the linear ruling approach. I have shown that the linear ruling approach does not always work. Conceivably there could be some other way to construct a developable surface. I don't think there is another way, but I haven't proved this.
On the positive side, here is a construction that often works. At a given point P on curve A, construct the tangent line, L. Take a plane containing this tangent line, and rotate it around L until it touches curve B (tangentially), say at a point Q. This may not be possible, but it often is. Take the line between P and Q as a ruling. The ruled surface constructed this way is developable.