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 -in a polygon.
Also of interest is the family of rectangles inscribed in smooth curves REF .
Rectangle Coincidences and Sweepouts
arXiv: Metric Geometry
Mathematics