  At any point of a surface in the four-dimensional Euclidean space we consider
the geometric configuration consisting of two figures: the tangent indicatrix,
which is a conic in the tangent plane, and the normal curvature ellipse. We
show that the basic geometric classes of surfaces in the four-dimensional
Euclidean space, determined by conditions on their invariants, can be
interpreted in terms of the properties of the two geometric figures. We give
some non-trivial examples of surfaces from the classes in consideration.
