Curve with arc length have signed curvature k(s)>0?

Question

Let $g:I \to \mathbb{R}^2$ be a curve such that for all $s \in I$, $\|g'(s)\|=1$ and $\kappa_g(s) \neq 0$, where $\kappa_g$ is the signed curvature of $g$. Is $\kappa_g(s) \gt 0$ for all $s \in I$? thanks!

Answer 1

If for all $s \in I$, $k(s) \neq 0$, then using IVT (as the signed curvature is a continuous function) either $k(s) > 0$ for all $s \in I$ or $k(s) < 0$ for all $s \in I$. We can't conclude that $k(s) > 0$ for all $s \in I$ because it is also possible that $k(s) < 0$ for all $s \in I$.

For instance, $t \in (0,2\pi) \mapsto (\cos t, -\sin t)$ has a strictly negative signed curvature everywhere.

The signed curvature depends on the orientation of the curve. From a curve with strictly positive signed curvature, you can extract a curve with strictly negative signed curvature just by changing its orientation.