Contact cylindrical surfaces and a projection of a surface around a parabolic point

We investigate differential geometric properties of a parabolic point of a surface in the Euclidean three space. We introduce the contact cylindrical surface which is a cylindrical surface having a degenerate contact type with the original surface at a parabolic point. Furthermore, we show that such a contact property gives a characterization to the $\mathcal{A}$-singularity of the orthogonal projection of a surface from the asymptotic direction.