This paper is dedicated to the study of metrical properties of a collection of 2D thinning algorithms that we have proposed. Here, we characterize their underlying metrics and use it to reduce the classical metrical biases that affect thinning algorithms in the square grid. We show that some algorithms from the collection lead to skeletons based on a particular geometry, corresponding to the (4,8)-median axis, which is a new shape descriptor, featuring nice robustness and conditioning properties.