Representation of Curves of Constant Width in the Hyperbolic Plane

Paulo Ventura Araújo

Abstract: If $\gamma$ is a curve of constant width in the hyperbolic plane $\H^{2}$, and $l$ is a diameter of $\gamma$, the {\sl track function} $x(\theta)$ gives the coordinate of the point of intersection $l(x(\theta))$ of $l$ with the diameter of $\gamma$ that makes an angle $\theta$ with $l$. We show that $x(\theta)$ determines the shape of $\gamma$ up to the choice of a constant; this provides a representation of all curves of constant width in $\H^{2}$. The track function is locally Lipschitz on $(0,\pi)$, satisfies $|x'(\theta)\sin\theta|<1-\epsilon$ for some $\epsilon>0$, and, if $l$ is appropriately chosen, has a continuous extension to $[0,\pi]$ such that $x(0)=x(\pi)$; conversely, any function satisfying these three conditions is the track function of some curve of constant width. As a by-product of the representation thus obtained, we prove that each curve of constant width in $\H^{2}$ can be uniformly approximated by real analytic curves of constant width, and extend to all curves of constant width some results previously established under restrictive smoothness assumptions.

Keywords: Constant width; hyperbolic plane.

Classification (MSC2000): 52A10, 51M10

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