Elliptic cylinder ($u^2/a^2+v^2/b^2=1$)

For $a,b>0$, an elliptic cylinder is a quadric surface with the displayed equation in suitable Cartesian coordinates, independent of the third coordinate. It is formed by translating an ellipse parallel to a line perpendicular to its plane. The generator direction is the zero-eigenvalue direction of its quadratic part, while the transverse semiaxes are $a,b$. It is circular if $a=b$.