You might see if this paper helps:
Ranestad, Kristian, and Bernd Sturmfels. "On the convex hull of a space curve." arXiv:0912.2986 (2009). (arXiv abstract link)
![CurveHull][1]
The yellow surface is $z - 4x^3 + 3x = 0$. The green surface has degree $16$. The pink triangle is planar.
(Image due to Frank Sottile, Philip Rostalski.)
If an approximate hull would suffice, it is "easy" to compute the 3D convex hull of many points along your curves.