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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.

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Concerning only "the task of constructing a 3D space curve from the planar (κ(s),τ(s)) trajectory," see this MSE question: One needs to solve a system of 1st-order DiffEqs (as you realize).

John Oprea has Maple code to actually solve these equations, at this link. He writes,

The following procedure solves the nine Frenet equations together with the three equations defining the unit tangent vector of the curve ...

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

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    $\begingroup$ thanks for mentioning the differential equations and for your nice illustration. My suggestions were aimed at demonstrating, that there is a realistic chance to approximate with low effort the curves I described. I think that most people (including me) have difficulties setting up the differential equations for space filling or fractal curves; hence my 'hands on' proposal. $\endgroup$ – Manfred Weis Jan 6 '14 at 19:15

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