Curve with constant torsion and no curvature

Question

When curvature and torsion are given a curve is fully defined (upto Euclidean motions) in 3-space.

$ k=const , \tau = 0 $ represents a circle in a plane ;

But what does the space curve

$$ k =0 , \tau= const,$$

represent?

The center line $ (u=0) $ of a right handed twisted helicoid with parametrization $( u \cos v, u \sin v, c \;v ) $ is a good example.

Curvature/Torsion of $u=0$ line of helicoid.

Clearly u=0 is a straight line at the helicoid mid with zero curvatures ( both normal (asymptotic $ k_n=0$) and geodesic $k_g=0$ ) as valid for a full straight line.

Using Enneper-Beltrami theorem torsion of the central parametric line at $\; u=0$ is found constant:

Evaluating Gauss curvature K

$$ K= \dfrac{-c^2}{(c^2+u^2)^2}, \tau = \sqrt{-K}= \pm \dfrac {1}{c}$$

The sign for the torsion of right helicoid is positive and, for the left handed helicoid it is negative.

A physical example is of a long human hair that can be twisted right or left with constant torsion even if the twist is not clearly visible. Other examples include long straight portions of DNA and other polymer molecules which inhabit such a surface.

EDIT1:

In another example the straight line parameterized by $$(x,y,z)= (a, b t, c t) $$ has zero curvature and non-zero torsion in this example when it becomes asymptotic on certain (arbitrary?) surfaces surfaces of negative Gauss curvature.

EDIT2:

What I meant by torsion without curvature is shown in the first figure Twist of Helicoid's straight/geodesic Spine. The special asymptotic line intrinsically characterizes how twist occurs during parallel transport in tangent spaces.

Answer 1

If a 'curve' $\gamma:\>s\mapsto{\bf r}(s)$ is parametrized with respect to arc length then by definition $\kappa(s):=|\ddot{\bf r}(s)|$. The assumption $\kappa(s)\equiv0$ then implies $\ddot{\bf r}(s)\equiv{\bf 0}$, and this leads to ${\bf r}(s)={\bf a}+s {\bf u}$, where ${\bf u}$ is a unit vector.

Now, Torsion measures the failure of a curve to be planar. If $\gamma$ has zero torsion, it lies in a plane. Hence for $\kappa=0 \implies \tau = 0$ corresponding to a line. Lines look very much like lines, and they are certainly planar.

Answer 2

The whole theory of curvature and torsion of "curved lines" is based on the implicit assumption that the curvature is not zero except for isolated points. It is zero on a segment if and only if the curve segment is a straight line.

As the Wikipedia article Torsion of a curve states:

Let $\bf{C}$ be a space curve parametrized by arc length $s$ and with the unit tangent vector $\bf{t}$. If the curvature $\kappa$ of $\bf{C}$ at a certain point is not zero then the principal normal vector and the binormal vector at that point are the unit vectors $$ \bf{n}=\frac{\bf{t}'}{\kappa}, \quad \bf{b}=\bf{t}\times\bf{n}, $$ where the prime denotes the derivative of the vector with respect to the parameter $s$. The torsion $\tau$ measures the speed of rotation of the binormal vector at the given point. If is found from the equation $$ \bf{b}' = \tau\,\bf{n}. $$

Note carefully that the definition of $\bf{n}$ involves dividing by the curvature. Hence, if the curvature is $0$, $\bf{n}$ is not defined. Because this vector is not defined, this implies that the torsion can not be defined either since its definition uses $\bf{n}$.

Of course, you could just pick any fixed unit vector $\bf{n}$ which is perpendicular to the tangent $\bf{t}$ and then define $\bf{b}$ the usual way which leads to a constant binormal and hence by the definition of torsion we find $\,\tau=0.\,$ Obviously, the normal vector $\,\bf{n}\,$ is not unique, but in any case torsion is forced to be $\,0.\,$

Answer 3

This is a late response. But I wanted to point out that this question is not entirely devoid of merit. These are subtle pathological cases that arise in modeling problems in kinematics, mechanics and curve design and therefore justifiably dismissed by mainstream mathematicians. The notion of a twisted straight line with zero curvature but non zero torsion is mentioned in Nutbourne and Martin's book "Differential geometry applied to curve and surface design". See this link here. I have a copy of the book. Maybe it would be unwise to use the Frenet frame to tackle this issue.

But as an aside, if one takes a circular helix of radius $r$ and pitch $p$, then as $r \to 0$, it is easy to observe that the curvature goes to $0$, but the torsion goes to $1/p$. Indeed the straight axis of the helicoid can probably be treated this way.