Drawing a geometric conclusion from the curvature and torsion of a curve

Question

If I was working with a curve $\tilde{c}(t)$ and found that the curvature $\displaystyle \kappa(t) = \frac{1}{8\sin(\frac{t}{2})}$ and the torsion $\tau(t) = 0$. What geometric conclusion, about the curve, should I be able to draw from that?

Answer 1

Firstly, the torsion being $0$ simply means that at every point on the curve, it is not bending away from the plane containing the curve, which means that the whole curve can be captured on a plane.

Now the curvature part is a bit suspicious, because whenever $t=2n \pi$, the denominator is $0$, and the curve has infinite curvature at the points. From the definition of curvature, usually we seldom encounter infinite curvature because that would usually imply that either $\tilde c ''(t) \to \infty$ or $\tilde c '(t) =0$. In the first case, the curve is not continuous, whereas for the second case, the curve is not regular. So my guess is that the curve is only defined on an interval between $2n \pi $ and $2(n+1) \pi$. Then the curve could have curvature of positive or negative signs, so it could be concave up or down. At the flattest point $(2n+1)\pi$, the curve has $\frac 18$ curvature, but it can also get arbitrarily curvy.