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.
If an approximate hull would suffice, it is "easy" to compute the 3D convex hull of many points along your curves.