On Curves and Surfaces of Constant Width

TL;DR

This paper explores properties of curves and surfaces of constant width, utilizing Fourier series to derive geometric bounds, packing densities, and constructing new examples of such surfaces.

Contribution

It introduces a Fourier series method for constructing new constant width surfaces and establishes bounds and properties related to curvature and packing density.

Findings

Perimeter of an oval equals π times its average width.

Provides a bound on the radius of curvature based on Fourier harmonics.

Density of a specific packing exceeds that of circles, suggesting a maximum for constant width curves.

Abstract

This paper focuses on curves and surfaces of constant width, with some additional results about general ovals. We emphasize the use of Fourier series to derive properties, some of which are known. Amongst other results, we show that the perimeter of an oval is $\pi$ times its average width, and provide a bound for the radius of curvature of an oval that depends on the structure of the harmonics in its Fourier series. We prove that the density of a certain packing of Reuleaux curved triangles in the plane is $$\frac{2 (\pi - \sqrt{3})}{\sqrt{15} + \sqrt{7} - 2 \sqrt{3}} \simeq 0.92288$$ which exceeds the maximum density for circles ($\simeq 0.9060$), and conjecture this is the maximum for any curve of constant width. For surfaces of constant width we show that $ \rho_0(P)+ \rho_1(Q)=w$ where the $\rho_i$ are the principal curvatures, $P$ and $Q$ are opposite points, and $w>0$ is the width. Moreover, an ovoid is a surface of constant width $w>0$ if and only if $\rho_{{\rm mean}}(P)+\rho_{{\rm mean}}(Q) = w$ where $\rho_{{\rm mean}}(P)$ is the average radius of curvature at point $P$. Finally, we provide a Fourier series-based construction that produces arbitrarily many new surfaces of constant width.