In ancient mathematics circular cylinders were the only ones studied, but since some of the properties are as easily proved for the case of a noncircular Directrix, it is not now customary to limit them in this way.
"The Teaching of Geometry"
David Eugene Smith
The Directrix is at right angles to the principal axis.
"An Elementary Course in Synthetic Projective Geometry"
Lehmer, Derrick Norman
In a parabola the vertex is equally distant from the focus and the Directrix, these three points and the point at infinity on the axis being four harmonic points.
"An Elementary Course in Synthetic Projective Geometry"
Lehmer, Derrick Norman