$\textbf{Problem 25.2.}$ ($\textit{Differential Geometry}$ by Erwin Kreyszig). Find a parametric representation of the cylinder generated by a straight line $G$ which moves along a curve $$C: \textbf{x}(s)= (h_{1}(s), h_{2}(s), h_{3}(s))$$ and is always parallel to the $z$-axis.
My attempt: The parametric representation of a cylinder that is always parallel to the $z$-axis is $$\textbf{F}(u,v)=(a\cos(v),a\sin(v),u)$$ where $a$ is the radius and $0\leq v \leq 2\pi$, $u$ is the height. Now our cylinder is generated by a straight line $G$ which is always parallel to the $z$-axis and moves along curve $\langle (h_{1}(s),h_{2}(s),h_{3}(s)\rangle$ so this means replacing $a$ with points along the curve and parallel to the $z$-axis. This is the part I am having a lot of trouble with conceptually. If the cylinder is generated by a line that moves along this curve, but is always parallel to the $z$-axis, then this means that points along the curve should be the radius, which could be calculated simply by $h_{1}(s)^{2}+h_{2}(s)^{2}=a^2$ and then $h_{3}(s)=u$ (in the original equation.) This gave me $$\textbf{F}(u,v)=\bigg(\sqrt{h_{1}(s)^{2}+h_{2}(s)^{2}}\cos(v), \sqrt{h_{1}(s)^{2}+h_{2}(s)^{2}}\sin(v), h_{3}(s)\bigg)$$.
I still feel uncertain about this answer. Please help me get back on the right track. Thank you for your time.
There's nothing to do with circles here. Just grab a line and move it parallel along the given curve. If the direction vector of the line is $\mathbf v$ and the curve is given parametrically by $\boldsymbol\alpha(s)$, then consider $\mathbf F(s,t)=\boldsymbol\alpha(s)+t\mathbf v$.