(a) Let $\alpha: I \rightarrow \mathbb{R}^{3}$ be a smooth regular curve, parametrized by arc length, such that $\alpha^{\prime \prime}(s) \neq 0$ for all $s \in I$. Define the Frenet frame associated to $\alpha$ and derive the Frenet formulae, identifying curvature and torsion.

(b) Let $\alpha, \tilde{\alpha}: I \rightarrow \mathbb{R}^{3}$ be as above such that $\tilde{k}(s)=k(s), \tilde{\tau}(s)=-\tau(s)$, where $k, \tilde{k}$ denote the curvature of $\alpha, \tilde{\alpha}$, respectively, and $\tau, \tilde{\tau}$ denote the torsion. Show that there exists a $T \in \mathrm{O}_{3}$ and $v \in \mathbb{R}^{3}$ such that

$$\alpha=T \circ \tilde{\alpha}+v$$

[You may appeal to standard facts about ordinary differential equations provided that they are clearly stated.]

(c) Let $\alpha: I \rightarrow \mathbb{R}^{2}$ be a closed regular plane curve, bounding a region $K$. Let $A(K)$ denote the area of $K$, and let $k(s)$ denote the signed curvature at $\alpha(s)$.

Show that there exists a point $s_{0} \in I$ such that

$$k\left(s_{0}\right) \leqslant \sqrt{\pi / A(K)}$$

[You may appeal to any standard theorem provided that it is clearly stated.]