  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.
