Right now, some of the most interesting work with 3-dimensional curve theory involves contact geometry/topology. In particular, J. Etnyre and P. Massot have published a couple of papers which explore the relationships between the contact topology of 3-dimensional contact manifolds and the resulting geometry of curves within those spaces. I know one question that they were exploring a while back was whether the existence of a closed curve with a certain amount of writhe/knottted-ness could determine whether the ambient 3-manifold had a tight contact structure or not.

A Poincare sphere describing polarization states is not symmetric.

Regularity of area-minimizing surfaces in 3-D Euclidean space.

We can study curves in a space-time, this is very interesting.

Also of interest is the relationship between geometry of curves and surfaces and integrable equations. You can look at the work of E. Ferapontov, C. Roger and J. Schiff, and K. Tenenblat

Find necessary and sufficient conditions on curvature and torsion, so that a curve in 3-dimensional euclidean space is closed. Note that periodicity is not enough!

Related to Brendan Foreman's answer: For the last 30+ years, I've been studying the curves in a round (or pseudoconvex) 3-sphere S in the complex affine place C^2 that are intersections of S with complex plane curves V_f = f^{-1}(0), where f(z,w) is holomorphic in (at least) some open ball containing S (for instance, f might be a polynomial in two complex variables). Call any such intersection G_f=S\cap V_f a C-link in S. The two special cases that have been most thoroughly investigated are "transverse C-links", where it's assumed that the intersection is transverse (so G is the disjoint union of finitely many naturally oriented smooth simple closed curves) and "totally-tangential C-links", where it's assumed that the exhaustion function defining S (e.g., |z|^2+|w|^2 if S is the round sphere centered at (0,0)) has Morse index 1 at each point of G (so G is the disjoint union of finitely UNoriented, but naturally framed, smooth simple closed curves). Totally-tangential C-links are exactly the same as real-analytic Legendrian links in their ambient contact 3-manifold (where the contact plane at each point is, by definition, the unique complex line in the tangent real 3-space of the 3-sphere). Transverse C-links are harder (in some sense; and maybe only because no one has seen how easy it is!) to characterize *exactly*. (1) Up to ambient isotopy (i.e., "knot type") they are characterized as being exactly the "quasipositive links" in a sense I defined. Here, an element of the braid group B_n is "quasipositive" if it is in the subsemigroup generated by any one of (and therefore all of) the standard generators, and a link is "quasipositive" if it can be presented as the closure of a quasipositive braid. It is elementary to show that a quasipositive link is a transverse C-link; Boileau and Orevkov proved the converse (which I had conjectured based mostly on sheer optimism) using symplectic topology (Eliashberg stuff). (2) As oriented smooth 1-chains in S^3, transverse C-links are (up to some technical details that I've forgotten ... ) exactly those links over which all complex-valued 1-forms of type (0,1) (more or less all linear combinations of d(z-bar) and d(w-bar)) have integral 0 (that these integrals are 0 is just Stokes's theorem applied to the piece of complex plane curve bounded inside S by the link; the converse is hard, or hardish, complex analysis by Wermer and others, using uniform algebras etc etc).

But there isn't any really good differential-geometric characterization of either class of C-links.

One idea I've been kicking around recently may be of interest to you. Look at the case S = S^3, the unit 3-sphere, and give the open unit ball its natural structure as the complex hyperbolic plane CH^2. Now, if G_f = V_f\cap S^3 is a transverse C-link, the "open piece of complex plane curve " W_f=V_f\cap CH^2 is (tautologously) a complex curve in CH^2. There are lots of complex curves in CH^2, but they haven't been much studied (except for ones that are stable under some interesting group of automorphisms of CH^2 that quotients out CH^2 itself to a complex-hyperbolic complex surface, containing the quotiented-out complex curve as a complex curve). W_f has, in a natural geometric sense, "a link at infinity in CH^2" (namely, G_f); plenty of complex curves W in CH^2 DON'T have any "link at infinity" (for instance, ones that are stable under an infinite group of automorphisms; or, more generally, any W of "infinite topological type", that is, which is not diffeomorphic to a subsurface of a compact Riemann surface). Can the differential geometries of CH^2 and S^3, alone or together, characterize transverse C-links in this role as "links at infinity"? I'm presently working (with great difficulty, since my differential-geometric chops are particularly weak) on a conjecture that a complex curve W in CH^2 of finite topological type has a link at infinity if and only if a single certain (curvature-related) integral over W is finite. Perhaps a similar approach could characterize transverse C-links by finiteness of a *single* differential-geometrically-defined integral (as contrasted with a countable set of moment integrals being 0, in Wermer's theorem)?

Getting back to totally-tangential C-links, of course Etnyre and many others have made good use of them (as such) already in their contact geometrical/topological investigations, but I'm sure there's still stuff to be done there (and naturally I encourage you to do it). For starters, no one (that I know of) has yet made Boileau and Orevkov's theorem constructive, in any of various senses--for instance, (i) supposing you have, concretely, a transverse C-link given as G_f, how can you find an explicit quasipositive braid with closure isotopic to G_f ? or even (ii) supposing you have a braid, how can you determine whether or not it's quasipositive?. These are also questions with an installed base of people already thinking about them (at least some of the time; for instance, Orevkov would love an algorithm for determining braid quasipositivity, because of applications to configurations of ovals in the real projective plane; see Gabard's recent survey on Hilbert's 16th Problem at the arxiv), but again that doesn't mean you might not be able to supply the piece they've missed!

Cheers, Lee Rudolph

Call the curve nondegenerate if the oscullating frame is nondegenerate.

Question: compute the number of connected components of the space of nondegenerate closed curves in a given 3d manifold.

It appears to be an open questions as to whether there are any closed curves of nonzero constant torsion embedded in a given convex surface. I recently published a paper that partially addresses this question (a preprint is also available on ArXiv):

http://link.springer.com/article/10.1007%2Fs00022-013-0166-2

The best that my coauthor and I could do was derive an upper bound for the torsion of such a curve that is expressed in terms of the principal curvatures of the surface.

Retrieving a 3D curve from two of the uncalibrated digital camera images

"Retrieving a 3D curve from two of the uncalibrated digital camera images".

I don't know about digital camera technology, so this may be a dumb question, but what information is in an "uncalibrated digital camera image" that is not in the orthogonal projection onto a plane? Presumably there's *something* more than that, because even a 3D line segment can't be retrieved (say, up to rigid motions) from any two of its projections!