arbelos and parbelos

The arbelos is the plane region bounded by three pairwise tangent semicircles with diameters on the same line.

The arbelos was known already in classical Greek geometry.  It has many interesting properties; see e.g. http://mathworld.wolfram.com/Arbelos.htmlMathworld.  One is that the distance between the two outermost points along the inner semicircles of the arbelos is the same as the distance along the outer semicircle, namely, its radius times $\pi$.

The parbelos, a parabolic analog of the arbelos, is theplane region bounded by the latus rectum (http://planetmath.org/Hyperbola) arcs of three parabolas with latera recta AB, BC, AC, where the points A, B, C lie on a line.  Unlike in the arbelos, the arcs of the parbelos are not pairwise tangent: the inner two are tangent to the outer one, but not to each other.  The parbelos has several interesting properties which can be seen in Sondow’s article [1]; see also Tsukerman’s paper [2].

Some of them are analogous to the properties of the arbelos.For example, the distance between the two outermost two pointsof the parbelos along the inner arcs is the same as along theouter arc, namely, its semilatus rectum times theuniversal parabolic constant (http://planetmath.org/ArcLengthOfParabola)$P$.