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 the plane 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 points of the parbelos along the inner arcs is the same as along the outer arc, namely, its semilatus rectum times the universal parabolic constant (http://planetmath.org/ArcLengthOfParabola) $P$.

References

• 1 Jonathan Sondow: The parbelos, a parabolic analog of the arbelos. – Amer. Math. Monthly 120 (2013) 929–935.  Also in http://arxiv.org/abs/1210.2279arXiv (2012).
• 2 Emmanuel Tsukerman: Solution of Sondow’s problem: a synthetic proof of the tangency property of the parbelos.  – Amer. Math. Monthly 121 (2014) 438–443. Also in http://arxiv.org/abs/1210.5580arXiv (2012).
 Title arbelos and parbelos Canonical name ArbelosAndParbelos Date of creation 2014-06-29 8:56:48 Last modified on 2014-06-29 8:56:48 Owner pahio (2872) Last modified by pahio (2872) Numerical id 19 Author pahio (2872) Entry type Topic Classification msc 53A04 Classification msc 53-03 Classification msc 51M99 Classification msc 01A20 Classification msc 00A08 Defines arbelos Defines parbelos