# Existence and Properties of 3D Curves with unusual 2D $(\kappa(s),\tau(s))$ Trajectories

This question is inspired by Surface in 3D that realizes all pairs of principal curvatures

While one can imagine, that a 3D surface could exist, that realizes all pairs of principal curvatures, because the set of all pairs of principal curvatures is, losely speaking, also 2D, it is counterintuitive that also a 3D curve could exists, that realizes all pairs of curvature $\kappa(s)$ and torsion $\tau(s)$ for $s \in (-\infty,+\infty)$.
However, interpreting $(\kappa(s),\tau(s))$ as a parametric curve in the Euclidean plane, it is fairly easy to conclude, that $(\kappa(s),\tau(s))$ must be space-filling.

Questions:

• what is the fractal dimension of 3D curves with space-filling $(\kappa(s),\tau(s))$ trajectory?

• what is the radius of the smallest 3D sphere into which such a 3D curve fits?

• what about selfintersection of the 3D curve?

• what about 3D curves, whose $(\kappa(s),\tau(s))$ trajectory has fractal dimension between $1$ and $2$, maybe resembling one of the prominent fractal 2D curves (cf e.g. http://en.wikipedia.org/wiki/Category:Fractal_curves)?

• have such 3D curves been described or studied?

Update

the task of constructing a 3D space curve from the planar $(\kappa(s),\tau(s))$ trajectory is an interesting problem itself; it seems possible to define such a reconstruction in a canonical form, namely that

• the frenet frame shall vary continously with $s$

• $s$ shall be a length parameterization of both the 2D trajectory and the associated 3D curves

Ideas for reconstruction the 3D curve:

• use a point sampling of the $(\kappa(s),\tau(s))$ trajectory, calculate the length of the trajectory between two adjacent sampling points and use that as the parameter range for a segment of a helix, whose curvature and torsion equal the coordinate values of the sample point. The desired 3D curve is then obtained by letting the number of sampling points tend to $\infty$. However, in that construction curvature and torsion are not continuous for the approximate 3D curves, but I'm not sure if that is an issue for the limit curve.

• use rectilinear approximations of the $(\kappa(s),\tau(s)))$ and smoothly piece together so called $Salkowski$ and $anti$-$Salkowski$ curves (cf e.g. http://www.uv.es/~monterde/pdfsarticlesmeus/CAGD-6.pdf); here I don't know, whether it can be guaranteed that $s$ simultaneously parameterizes the 2D trajectory and the associated 3D curve by length. This construction would fit Hilbert curves as spacefilling planar trajectories.

• for piecewise linear approximations of the trajectory use segments of 3D curves for which both curvature and torsion are linear functions of $s$; however, I could not find a description of such curves.

I hope, that the above suggestions help in visualizing such strange curves and, at least approximately, allow answering other questions related to such strange curves.

For example, for $\kappa(t)=t$ and $\tau(t)=t/10$: