Elliptic cylinder

A cylindrical surface of the second order having an ellipse as directrix. If this ellipse is real, then the surface is called real and its canonical equation has the form \begin{equation} \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1; \end{equation}

if the ellipse is imaginary, then the surface is called imaginary and its canonical equation has the form \begin{equation} \frac{x^2}{a^2} + \frac{y^2}{b^2} = -1. \end{equation}