Local scales on curves and surfaces

TL;DR

This paper extends the concept of local scales from functions to curves and surfaces, analyzing deviations from linear or planar approximations at multiple scales, and applying singular integral theory to establish their properties.

Contribution

It introduces a new framework for defining and analyzing local scales on curves and surfaces, generalizing previous work on functions, and demonstrates their invariance under dilation.

Findings

Local scales measure deviations from linear or planar structures at various scales.

The theory of singular integral operators helps establish properties of local scales.

Local scales are invariant under dilation.

Abstract

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.