Digital planes are sets of integer points located between two parallel planes. We present a new algorithm that computes the normal vector of a digital plane given only a predicate “is a point x in the digital plane or not”. In opposition with the algorithm presented in [7], the algorithm is fully local and does not explore the plane. Furthermore its worst-case complexity bound is , where is the arithmetic thickness of the digital plane. Its only restriction is that the algorithm must start just below a Bezout point of the plane in order to return the exact normal vector. In practice, our algorithm performs much better than the theoretical bound, with an average behavior close to . We show further how this algorithm can be used to analyze the geometry of arbitrary digital surfaces, by computing normals and identifying convex, concave or saddle parts of the surface.