Rectangle Coincidences and Sweepouts

TL;DR

This paper establishes an integral formula for inscribed rectangles in loops and demonstrates that the count of rectangle coincidences increases linearly with the number of diameters of the polygon, under mild conditions.

Contribution

It introduces a new integral formula for continuous rectangle paths and links rectangle coincidences to polygon diameters, revealing a linear growth relationship.

Findings

Number of rectangle coincidences grows linearly with polygon diameters

Integral formula for inscribed rectangles in loops

Relationship holds under mild genericity conditions

Abstract

We prove an integral formula for continuous paths of rectangles inscribed in a piecewise smooth loop. We then use this integral formula to show that (with a very mild genericity hypothesis) the number of rectangle coincidences, informally described as the number of inscribed rectangles minus the number of isometry classes of inscribed rectangles, grows linearly with the number of positively oriented extremal chords -- a.k.a. diameters -- of the polygon