We extend the study of local scales of a function (image) f to studying local scales on curves and surfaces G; in particular, we consider G to be a d-dimensional surface. In the case of a function, the local scales of f at x are computed by measuring the convolution of f with a kernel of zero mean and zero first moments of various scales. From the goodness of fit point of view, the computation can be viewed as measuring deviations of f from a linear function near x at different scales t's. In the case of a d-dimensional surface G, the analogy is to measure deviations of G from a d-plane near x on G at various scale t's. This analogy is realized through convolving the measure restricted to G with a kernel of zero mean and zero first moments. Characterizations and examples of local scales will also be presented. These are joint work with P. Jones and F. Memoli.