  In this paper, we extend our previous work on the study of local scales of a
function to studying local scales on curves and surfaces. In the case of a
function f, the local scales of f at x is computed by measuring the deviation
of f from a linear function near x at different scales t's. In the case of a
d-dimensional surface E, the analogy is to measure the deviation of E from a
d-plane near x on E at various scale t's. We then apply the theory of singular
integral operators on E to show useful properties of local scales. We will also
show that the defined local scales are consistent in the sense that the number
of local scales are invariant under dilation.
