Equilibria of plane convex bodies

TL;DR

This paper derives a formula linking the number of horizontal equilibria of a planar convex body to the winding number of its evolute, extending classical results to cases where the center lies on the evolute.

Contribution

It introduces a new formula connecting equilibria counts with evolute winding numbers, including cases where the center is on the evolute.

Findings

Derived a formula for horizontal equilibria using evolute winding numbers

Extended the formula to cases where the center is on the evolute

Provided a modified formula for non-horizontal equilibria

Abstract

We obtain a formula for the number of horizontal equilibria of a planar convex body $K$ with respect to a center of mass $O$ in terms of the winding number of the evolute of $\partial K$ with respect to $O$. The formula extends to the case where $O$ lies on the evolute of $\partial K$ and a suitably modified version holds true for non-horizontal equilibria.