Determining and analyzing geometric features is often an essential step in understanding and classifying three-dimensional objects. Among geometric features, the normal vector field provides a local and global approach to the surface structure of objects. However, for digital surfaces, obtaining this field often requires parameter adaptation or comes up against configurations that do not exist in digital planes. In this sense, the normal vector field is different from the vector field defined by pointwise tangent planes. In this article, we propose an approach based on a fan of digital planes around a point. This allows us to robustly capture all local configurations and to adapt to the local flatness of digital surfaces. Experimental evaluations using multigrid approaches show that it is both faster and more robust than state-of-the-art methods in the field, while maintaining comparable accuracy.