calculate the constant radius of curvature of the curve $r(u)= (\cos(u), \sin(u), u )$

Question

I have calculated the velocity vector to have length $\sqrt{2}$.

Thus the curvature is $\frac{1}{ \sqrt2}$.

Does this mean the constant radius of curvature is then $\frac{1}{ \sqrt2}$?

Answer 1

The radius of curvature, $R$, at a point is the inverse of the curvature $K$ of the curve at this point: $$R=\frac{1}{K}.$$

The curvature should be $$K(u)=\frac{||\vec{r}'(u)\times \vec{r}''(u)||}{(||\vec{r}'(u)||)^{3}}=\frac{\sqrt{2}}{2\sqrt{2}}=\frac{1}{2}.$$