The $f$-belos

TL;DR

This paper generalizes the classical arbelos and parbelos shapes to a broader class called $f$-belos, establishing analogous properties and characterizations for these generalized figures.

Contribution

It introduces the $f$-belos as a new generalization of arbelos and parbelos, extending properties and providing characterizations of these shapes.

Findings

Established properties of $f$-belos analogous to arbelos and parbelos.

Characterized arbelos and parbelos within the $f$-belos framework.

Extended geometric understanding of shapes bounded by similar curves.

Abstract

The \emph{arbelos} is the shape bounded by three mutually tangent semicircles with collinear diameters. Recently, Sondow introduced the parabolic analog, the \emph{parbelos} and proved several properties of the parbelos similar to properties of the arbelos. In this paper we give one step further and generalize the situation considering the figure bounded by (quite) arbitrary similar curves, the \emph{$f$-belos}. We prove analog properties to those of the arbelos and parbelos and, moreover, we characterize the parbelos and the arbelos as the $f$-beloses satisfying certain conditions.