Ovals of Constant Width in Polar Coordinates (Contemporary Mathematics, 2025)
We explore ovals of constant width in polar coordinates in this paper. Conversion of a parametric function defined on a rectangular domain of angles, to a polar representation defined on a domain of polar angles is introduced, and the relationship between the rectangular angles and the polar angles is discussed. The length of the parametric curve in polar coordinates between opposite points and from one vertex point to the next can be determined using the oval’s vertices. A new verification of Barbier's theorem in polar coordinates is presented. We show that the extreme values of the radial coordinate of the discussed polar oval are obtained at both its vertices and opposite points. Ovals and specific circles with the origin at the center are compared, and we demonstrate that every given oval is analytically and geometrically enclosed between those two specific circles. Intersection points between a polar oval and any circle related to it, centered at the origin, are formulated. Simulation and numerical examples are presented to support the analytical and theoretical results.
Keywords: ovals; convex curve; constant width; support function; barbier's theorem; polar coordinates