I What Is the Geometrical Interpretation of Bounded Curves?

[wrobel]

It is well known that a curve in $\mathbb{R}^3$ is uniquely (up to a position in the space) defined by its curvature $\kappa(s)$ and torsion $\tau(s)$, here $s$ is the arc-length parameter. We will consider $\kappa(s),\tau(s)\in C[0,\infty)$

Thus a natural problem arises: to restore features of the curve from given functions $\kappa(s),\tau(s)$. This is not a simple problem: the Frenet-Serret equations are not in general integrable explicitly.A simplest property of the curve is boundedness. We shall say that a curve is bounded iff its radius-vector is bounded: $$\sup_{s\ge 0}|\boldsymbol r(s)|<\infty.$$

For example, if a function $\kappa(s)/\tau(s)$ is monotone and
$$\lim_{s\to\infty}\frac{\kappa(s)}{s\cdot\tau(s)}=0$$ then the curve is unbounded. This is an almost trivial fact, it follows from some another almost trivial theorem, for details see http://www.ma.utexas.edu/mp_arc/c/16/16-63.pdf

The comments are welcome. Particularly is there a geometrical interpretation of brought above proposition

 

[shina]

It is well known that a curve in $\mathbb{R}^3$ is uniquely (up to a position in the space) defined by its curvature $\kappa(s)$ and torsion $\tau(s)$, here $s$ is the arc-length parameter. We will consider $\kappa(s),\tau(s)\in C[0,\infty)$

Thus a natural problem arises: to restore features of the curve from given functions $\kappa(s),\tau(s)$. This is not a simple problem: the Frenet-Serret equations are not in general integrable explicitly.A simplest property of the curve is boundedness. We shall say that a curve is bounded iff its radius-vector is bounded: $$\sup_{s\ge 0}|\boldsymbol r(s)|<\infty.$$

For example, if a function $\kappa(s)/\tau(s)$ is monotone and
$$\lim_{s\to\infty}\frac{\kappa(s)}{s\cdot\tau(s)}=0$$ then the curve is unbounded. This is an almost trivial fact, it follows from some another almost trivial theorem, for details see http://www.ma.utexas.edu/mp_arc/c/16/16-63.pdf

The comments are welcome. Particularly is there a geometrical interpretation of brought above proposition

I find it amazing. But will you explain me the restore features of curve. Actually I am going to read about curves in detail but I need little guidance. Can you name some of the books which I can buy.

 

[shina]

I was saying that why you have used word bounded curves. Will you tell me its precise meaning.

 

[shina]

Be more specific what you are saying

 

[wrobel]

was saying that why you have used word bounded curves. Will you tell me its precise meaning.

I have already done that