# equation of a rotated circle

In 3d space i have a circle in the $xz$ plane defined by equation $(x-i)^2+(z-k)^2=r^2$. If this circle is rotated through $\theta$ degrees around the line $z=k$ or by $\phi$ degrees around the line $x=i$ or both. What now is the equation?

For a rotation around $z=k$, the resulting circle is the intersection between the sphere $(x-i)^2+(z-k)^2+y^2=r^2$ and the plane $y=\tan\theta(z-k)$.
• does it also hold for $x=i$ is the intersection of sphere and plane $y=tan\phi(i-x)$ Jul 20 '16 at 4:50
• is there any simple way to combine both $\theta$ and $\phi$ into one equation for a plane Jul 20 '16 at 19:03
• something like $y=tan \theta(z-k)+tan \phi(x-i)$ Jul 20 '16 at 19:39