PORTUGALIAE MATHEMATICA Vol. 55, No. 3, pp. 355372 (1998) 

Representation of Curves of Constant Width in the Hyperbolic PlanePaulo Ventura AraújoCentro de Matemática, Faculdade de Ciências do Porto, 4050 Porto  PORTUGALEmail: paraujo@fc.up.pt 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 byproduct 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 Full text of the article:
Electronic version published on: 29 Mar 2001. This page was last modified: 27 Nov 2007.
© 1998 Sociedade Portuguesa de Matemática
