In this paper we prove that the function giving the frequency of a class of patterns of digital planes with respect to the slopes of the plane is continuous and piecewise affine, moreover the regions of affinity are precised. It allows to prove some combinatorial properties of a class of patterns called (
)-cubes. This study has also some consequences on local estimators of area: we prove that the local estimators restricted to regions of plane never converge to the exact area when the resolution tends to zero for almost all slope of plane. Actually all the results of this paper can be generalized for the regions of hyperplanes for any dimension
The proofs of some results used in this article are contained in the extended version of this paper .
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