Spherical surface $\mathbf S : x^2 + y^2 + z^2 =1 ~$; Cylindrical suface $ \mathbf C : x^2 + (y-0.5)^2 = 0.25 $
Let $~ \mathbf S \cap \mathbf C = \mathbf K (t)$. Then $ \mathbf K(t) = \left[ t \sqrt {1-t^2} ~~~~ 1-t^2 ~~~~ t \right]^\mathsf T . $ Actually $ \mathbf K$ parameterizes only half of $\mathbf S \cap \mathbf C $, but for now, that is not important. The goal is to express $\mathbf K$ in cylindrical coordinates. The cylindrical coordinates $\mathbf r (\mathbf x) $ corresponding to the cartesian coordinates $\mathbf x$ are generally $$ \mathbf r(\mathbf x) = \left[\rho(\mathbf x) ~~~~ \phi(\mathbf x) ~~~~z(\mathbf x) \right]^\mathsf T = \left [ \sqrt{x^2 + y^2} ~~~ \arctan \frac{y}{x} ~~~ z \right ]^\mathsf T .$$ Therefore $$\mathbf r (\mathbf K (t)) = \left [ \sqrt{1- t^2} ~~~~ \arctan \frac{\sqrt{1-t^2}}{t} ~~~~ t \right ]^\mathsf T . $$
Also: $\hat {\mathbf h}_\rho = \left[ \cos \phi ~~ \sin \phi ~~ 0 \right]^\mathsf T$ ; $\hat {\mathbf h}_\phi = \left[ -\sin \phi ~~ \cos \phi ~~ 0 \right]^\mathsf T $ ; $\hat {\mathbf h}_z = \left[ 0 ~~ 0 ~~ 1 \right]^\mathsf T $
$$\cos (\phi (\mathbf K (t))) = \cos \left(\arctan \left( \frac{\sqrt{1-t^2}}{t} \right) \right) = \frac{1}{\sqrt{1+ \left( \frac{\sqrt{1-t^2}}{t} \right)^2}} = t $$
$$\sin (\phi (\mathbf K (t))) = \frac{\sqrt{1-t^2}}{t} ~~ \cos \left(\arctan \left( \frac{\sqrt{1-t^2}}{t} \right) \right) = \sqrt{1-t^2}$$
Then $$ \left[ \begin{matrix} \hat {\mathbf h}_\rho & \hat {\mathbf h}_\phi & \hat {\mathbf h}_z \\ \end{matrix} \right]_{\mathbf r (\mathbf K(t))} = \left[ \begin{matrix} t & -\sqrt{1-t^2} & 0 \\ \sqrt{1-t^2} & t & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right] $$
The curve $\mathbf K$ expressed in cylindrical coordinates therefore is: $$\mathbf K (t) = \left[ \begin{matrix} \hat {\mathbf h}_\rho & \hat {\mathbf h}_\phi & \hat {\mathbf h}_z \\ \end{matrix} \right] \left[ \begin{matrix} t & \sqrt{1-t^2} & 0 \\ -\sqrt{1-t^2} & t & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right] \left[ \begin{matrix} \sqrt{1- t^2} \\ \arctan \frac{\sqrt{1-t^2}}{t} \\ t \\ \end{matrix} \right ] = \left[ \begin{matrix} \hat {\mathbf h}_\rho & \hat {\mathbf h}_\phi & \hat {\mathbf h}_z \\ \end{matrix} \right] \left[\begin{matrix} f_1(t) \\ g(t) \\ f_2 (t) \end{matrix} \right].$$
Now here is the problem: $g(t) = t^2 + t \arctan \left ( \frac{\sqrt{1-t^2}}{t} \right) -1 \neq 0$. I think that the coefficient of $\hat {\mathbf h}_\phi$, in the correct expression of $~ \mathbf S \cap \mathbf C $, in cylindrical coordinates, is necessarily zero. If somebody can tell me where I made an error that caused the result $g(t) \neq 0$, or if they can prove or otherwise convince me that $g(t)$ is not necessarily zero, then I will set this question to a resolved state.
As stated in the original post, the goal is to express $\mathbf K$ in cylindrical coordinates. $$\mathbf K(t) = \left[ t \sqrt {1-t^2} ~~~~ 1-t^2 ~~~~ t \right]^\mathsf T ~~;~~ -1 \le t \le 1 $$
$$\mathbf r(\mathbf x) = \left(\rho(\mathbf x) ~~~~ \phi(\mathbf x) ~~~~z(\mathbf x) \right) = \left ( \sqrt{x^2 + y^2} ~~~ \arctan \frac{y}{x} ~~~ z \right ) .$$
$$\mathbf r (\mathbf K (t)) = \left ( \sqrt{1- t^2} ~~~~ \arctan \frac{\sqrt{1-t^2}}{t} ~~~~ t \right ) .$$
$$\left[ \begin{matrix} \hat {\mathbf h}_\rho & \hat {\mathbf h}_\phi & \hat {\mathbf h}_z \\ \end{matrix} \right]_{\mathbf r (\mathbf K(t))} = \left[ \begin{matrix} t & -\sqrt{1-t^2} & 0 \\ \sqrt{1-t^2} & t & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right]$$
The following statement is false. $$\mathbf K (t) = \left[ \begin{matrix} \hat {\mathbf h}_\rho & \hat {\mathbf h}_\phi & \hat {\mathbf h}_z \\ \end{matrix} \right] \left[ \begin{matrix} t & \sqrt{1-t^2} & 0 \\ -\sqrt{1-t^2} & t & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right] \left[ \begin{matrix} \sqrt{1- t^2} \\ \arctan \frac{\sqrt{1-t^2}}{t} \\ t \\ \end{matrix} \right ].$$
The following statement is true. $$\mathbf K (t) = \left[ \begin{matrix} \hat {\mathbf h}_\rho & \hat {\mathbf h}_\phi & \hat {\mathbf h}_z \\ \end{matrix} \right] \left[ \begin{matrix} t & \sqrt{1-t^2} & 0 \\ -\sqrt{1-t^2} & t & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right] \left[ \begin{matrix} t \sqrt {1-t^2} \\ 1-t^2 \\ t \end{matrix} \right] . $$
$\mathbf K$ expressed in cylindrical coordinates is $$ \mathbf K (t) = \left[ \begin{matrix} \hat {\mathbf h}_\rho & \hat {\mathbf h}_\phi & \hat {\mathbf h}_z \\ \end{matrix} \right] \left[ \begin{matrix} \sqrt{1-t^2} \\ 0 \\ t \\ \end{matrix} \right] .$$
QED