In this article we develop in the case of triangulated meshes the notion of normal cycle as a dissimilarity measure introduced in [13]. Our construction is based on the definition of kernel metrics on the space of normal cycles which take explicit expressions in a discrete setting. We derive the computational setting for discrete surfaces, using the Large Deformation Diffeomorphic Metric Mapping framework as model for deformations. We present experiments on real data and compare with the varifolds approach.