I have the following sphere and cylinder, respectively:
$\begin{align} x^2+y^2+z^2&=(2R)^2,\\ (x-R)^2+y^2&=R^2,\qquad R>0. \end{align}$
How can I parametrize the space curve formed by their intersection?
I have the following sphere and cylinder, respectively:
$\begin{align} x^2+y^2+z^2&=(2R)^2,\\ (x-R)^2+y^2&=R^2,\qquad R>0. \end{align}$
How can I parametrize the space curve formed by their intersection?
Equation (2) gives $y^2=R^2-(x-R)^2$. Substitute it in Equation (1): $ z^2=4R^2-(R^2-(x-R)^2)-x^2=4R^2-2Rx.$ So $x=(4R^2-z^2)/2R.$ Going back to (2) you get the equation $y^2=z^2(1-\frac{z^2}{4R^2}).$
If you replace $x, y, z$ by their quotients by $2R$, then the equations have better look $x=1-z^2, \quad y^2=z^2(1-z^2).$