  Vortices (flows with closed elliptic streamlines) are exact nonlinear
solutions to the compressible Euler equation. In this contribution, we use
differential geometry to derive the transformations between Cartesian and
elliptic coordinates, and show that in elliptic coordinates a constant
vorticity flow reduces to $\dot{\mu}=0$ and $\dot{\nu}={\rm const}$ along the
streamline $\mu_0$ that matches the vortex eccentricity.
