(a) Let $\alpha:(a, b) \rightarrow \mathbb{R}^{3}$ be a smooth regular curve parametrised by arclength. For $s \in(a, b)$, define the curvature $k(s)$ and (where defined) the torsion $\tau(s)$ of $\alpha$. What condition must be satisfied in order for the torsion to be defined? Derive the Frenet equations.

(b) If $\tau(s)$ is defined and equal to 0 for all $s \in(a, b)$, show that $\alpha$ lies in a plane.

(c) State the fundamental theorem for regular curves in $\mathbb{R}^{3}$, giving necessary and sufficient conditions for when curves $\alpha(s)$ and $\widetilde{\alpha}(s)$ are related by a proper Euclidean motion.

(d) Now suppose that $\widetilde{\alpha}:(a, b) \rightarrow \mathbb{R}^{3}$ is another smooth regular curve parametrised by arclength, and that $\widetilde{k}(s)$ and $\tilde{\tau}(s)$ are its curvature and torsion. Determine whether the following statements are true or false. Justify your answer in each case.

(i) If $\tau(s)=0$ whenever it is defined, then $\alpha$ lies in a plane.

(ii) If $\tau(s)$ is defined and equal to 0 for all but one value of $s$ in $(a, b)$, then $\alpha$ lies in a plane.

(iii) If $k(s)=\tilde{k}(s)$ for all $s, \tau(s)$ and $\tilde{\tau}(s)$ are defined for all $s \neq s_{0}$, and $\tau(s)=\tilde{\tau}(s)$ for all $s \neq s_{0}$, then $\alpha$ and $\widetilde{\alpha}$are related by a rigid motion.