This report describes the calculation of local errors in Chamfer masks both in two- and in three-dimensional anisotropic spaces. For these errors, closed forms are given that can be related to the Chamfer mask geometry. Thanks to these calculation, it can be obsrved that the usual Chamfer masks (i.e. 3x3x3 or 5x5x5) have an inhomogeneously distributed error. Moreover, it allows us to design dedicated Chamfer masks by controlling either the complexity of the computation of the distance map (or equivalently the number of vectors in the mask), or the error of the mask in $\mathbbZ^2$ or in $\mathbbZ^3$. Last, since Chamfer distances are usually computed with integer weights (and approximate the Euclidean distance up to a multiplicative factor), we demonstrate that the knowledge of the local errors allows a very efficient computation of these weights.