Parametrization of an intersection Hello I'm trying to parametrize the intersection of the sphere of centre $(0,0,0)$ and radius $R$ and the cylinder of centre $(R/2,0,0)$ and radius $R/2$. (the intersection is the surface)
Any hints of how to proceed? My guess is to switch into cylinder coordinates but I can't figure out what value I have to put for my $z$ coordinate. I don't understand the concept of parametrization very well, any comment is welcome.
 A: Given your equations:
$${x^2} + {y^2} + {z^2} = {R^2}$$
$${\left( {x - \frac{R}{2}} \right)^2} + {y^2} = {\left( {\frac{R}{2}} \right)^2}$$
Indroducing:
$$\begin{gathered}
  x = \frac{R}{2}(\cos (\varphi ) + 1) \hfill \\
  y = \frac{R}{2}\sin (\varphi ) \hfill \\ 
\end{gathered} $$
the second equation is satisfied.
From first equation now you get:
$$\begin{gathered}
  {\left( {\frac{R}{2}(\cos (\varphi ) + 1)} \right)^2} + {\left( {\frac{R}{2}\sin (\varphi )} \right)^2} + {z^2} = {R^2} \hfill \\
  {\left( {\frac{R}{2}} \right)^2}({\cos ^2}(\varphi ) + 2cos(\varphi ) + 1) + {\left( {\frac{R}{2}} \right)^2}{\sin ^2}(\varphi ) + {z^2} = {R^2} \hfill \\
  {\left( {\frac{R}{2}} \right)^2}{\sin ^2}(\varphi ) + {\left( {\frac{R}{2}} \right)^2}{\cos ^2}(\varphi ) + 2{\left( {\frac{R}{2}} \right)^2}cos(\varphi ) + {\left( {\frac{R}{2}} \right)^2} + {z^2} = {R^2} \hfill \\
  2{\left( {\frac{R}{2}} \right)^2}(cos(\varphi ) + 1) + {z^2} = {R^2} \hfill \\
  {z^2} = {R^2} - 2{\left( {\frac{R}{2}} \right)^2}(cos(\varphi ) + 1) \hfill \\
   \hfill \\
   \hfill \\ 
\end{gathered}$$
With $0 \leqslant \varphi  < 2\pi $
$$\begin{gathered}
  x = \frac{R}{2}(\cos (\varphi ) + 1) \hfill \\
  y = \frac{R}{2}\sin (\varphi ) \hfill \\
  z =  \pm \frac{{R\sqrt {1-\cos (\varphi )} }}{{\sqrt 2 }} \hfill \\ 
\end{gathered}$$
is given the parametrization, that describes two surfaces bounded by magenta coloured curve. We used cylindric parametrization.
For $R=1$ picture shows upper half.

