A sharp bound on the number of self-intersections of a trigonometric curve

TL;DR

This paper establishes a precise upper limit on the self-intersections of closed trigonometric curves and demonstrates that most such curves are Whitney-normal, enhancing understanding of their geometric properties.

Contribution

It provides a sharp bound on self-intersections and proves generic curves of this type are Whitney-normal, advancing geometric analysis of trigonometric curves.

Findings

Sharp bound on self-intersections of trigonometric curves

Most generic curves are Whitney-normal

Improved understanding of geometric properties of these curves

Abstract

We obtain a sharp bound on the number of self-intersections of a closed planar curve with trigonometric parameterization. Moreover, we show that a generic curve of this form is normal in the sense of Whitney.