Circle is to cylinder as ellipse is to what?

What do you call the geometric shape obtained when you give an ellipse thickness? Or in other words, a circle is to a cylinder as an ellipse is to what?

A circle is to a cylinder as an ellipse is to a cylinder. As noted in wiktionary, a cylinder is

A surface created by projecting a closed two-dimensional curve along an axis intersecting the plane of the curve.

When the two-dimensional curve is a circle, the cylinder is called a circular cylinder. When the axis is perpendicular to the plane of the curve, the cylinder is called a right cylinder. In non-mathematical usage, both right and circular are usually implied.

You might say that a circle is to a right circular cylinder as an ellipse is to a right elliptical cylinder.

Ellipsoid is of course not the answer. A circle is to a sphere as an ellipse is to an ellipsoid.