Take the Viviani curve intersection of a sphere and a cylinder. Analytically explain what happens to the intersection curve if you keep the radius of the sphere constant, fix one side of the cylinder at the origin, and change the radius of the cylinder. Especially, discuss the cases when the radius of the cylinder goes to $0$ and $\infty$.
It was intuitively quite clear to me that if you keep $R$ (radius of the sphere) constant, and vary $r$ (radius of the cylinder), then the intersection curve tends to two points when $r\to 0$ and it tends to the longitudinal great circle when $r\to \infty$.
But, I had to solve it analytically, and so, I wrote the equations
$$x^2+y^2+z^2=R^2\\x^2+y^2=rx$$
Substituting the value of $x^2+y^2$ from the second equation into the first, we get
$$z^2=R^2-rx$$
which looks like a parabola on the $ZX-$plane!!!
Am I going wrong anywhere? If not, then how can I proceed from here?
Best Answer
If $r=0$, $x=y=0$ and $z=\pm R$.
And if $r\to\infty$, as $x=\dfrac{x^2+y^2}r\le\dfrac{R^2}{r}\to 0$ the curve tends to the circle $x=0,y^2+z^2=R^2$.
The projection of the intersection curve on the plane $xz$ is inscribed on a parabola (one or two arcs).
https://en.wikipedia.org/wiki/Viviani%27s_curve#/media/File:Viviani-fenster-xyz.svg