We call a coordinate system $(u, v)$ orthogonal if its coordinate curves (the two families of curves $u=$ constant and $v=$ constant ) are orthogonal trajectories (for example, a Cartesian coordinate system or a polar coordinate system). Let $(u, v)$ be orthogonal coordinates, where $u=x^{2}+2 y^{2},$ and $x$ and $y$ are Cartesian coordinates. Find the Cartesian equation of the $v$ -coordinate curves, and sketch the $(u, v)$ coordinate system.