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

Geometric interpretation of the invariants of a surface in R^4 via the tangent indicatrix and the normal curvature ellipse