Unwinding spirals

TL;DR

This paper proves that certain spirals with sub-exponential decay cannot be transformed into straight lines via bi-Lipschitz maps, highlighting the geometric rigidity of these structures.

Contribution

It establishes a sharp mathematical result showing the impossibility of unwinding specific spirals into straight lines through bi-Lipschitz homeomorphisms.

Findings

No bi-Lipschitz map can unwind sub-exponential spirals into straight lines.

Logarithmic spirals can be unwound, demonstrating the sharpness of the main result.

The result clarifies the geometric limitations of bi-Lipschitz transformations on spiral structures.

Abstract

We show that there is no bi-Lipschitz homeomorphism of $\mathbb{R}^2$ that maps a spiral with a sub-exponential decay of winding radii to an unwinded arc. This result is sharp as shows an example of a logarithmic spiral.