# Curvature and Torsion of a Curve

I am currently working on a problem and think I know the answer but need verification. The question reads:

Let there be a curve with non-zero curvature and zero torsion. Show this curve is planar. If the curve is allowed zero curvature at one point, does this above statement still hold?

I have shown that the curve is planar with non-zero curvature and zero torsion. But when the curve has zero curvature $\textit{and}$ zero torsion, isn't the curve a straight line there? And if so, doesn't this straight line remain in the original plane normal to the constant $\boldsymbol b$?

• What does it mean for a curve "to be a straight line a particular point"? – Travis Mar 8 '17 at 0:33
• Doesn't it just mean that it remains straight and doesnt curve at all? – Felicio Grande Mar 8 '17 at 0:48
• The point is that the notion isn't well-defined. – Travis Mar 8 '17 at 0:51
• So I can just say that it is impossible to have a straight line at just one particular point, therefore the curve is not planar? – Felicio Grande Mar 8 '17 at 1:02
• The curve certainly can be planar. For example, the graph of $x \mapsto x^3$ is planar but its curvature vanishes at exactly one point. – Travis Mar 8 '17 at 1:03

## 1 Answer

Let $$f(t)=\begin{cases}(t,t^3,0)&t\leq 0\\(t,0,t^3)&t\geq 0\end{cases}$$ Here $f$ has zero torsion but is not planar. It has zero curvature only at $t=0$.

• Yes I am aware, but I am wondering what happens to the curve if it is allowed zero curvature at one single point. – Felicio Grande Mar 8 '17 at 1:22
• Oh, then I totally change my answer – Rene Schipperus Mar 8 '17 at 1:35
• Though to be honest I am not sure how the torsion is defined at $t=0$. – Rene Schipperus Mar 8 '17 at 1:36
• Ah I see. Is there a general theorem or a statement I can use to simply state that if it has zero curvature at a point it is not planar? – Felicio Grande Mar 8 '17 at 1:41
• @Travis Yes, I was wondering this also, is the intention of the problem that curvature is zero at some point and thus that torsion is undefined there ? How are the normal and binormal at such a point to be defined ? I think you have to specify these things before attempting to find a better counterexample or a proof. – Rene Schipperus Mar 8 '17 at 13:57