Closed Form for Half-Area Overlap Offset of 2 Unit Disks

TL;DR

The paper addresses the problem of determining the center separation between two unit disks that yields half of each disk's area in overlap. It develops a closed-form representation by linking the lens-area condition to a Kepler-type equation and leveraging beta-function identities, culminating in expressions involving the inverse regularized beta function and the Kepler E function. The central result is the exact closed form $D_{DHA}=2\sqrt{ I^{-1}_{\frac{1}{2}}(\tfrac{1}{2},\tfrac{3}{2}) }$, with equivalent forms using $\operatorname{archav}$ and $\mathbf{E}(-1,\cdot)$, and a precise numerical value $D_{DHA}\approx 0.8079455066$. This provides a symbolic, closed-form handle on a problem rooted in planar geometry, with potential extensions to related radius configurations and overlap criteria.

Abstract

The separation between the centers of two unit circles such that their overlapping area is exactly half of each's area is known to be around $0.8079455\dots$ (OEIS A133741). However, no closed form of this number is known. Here, we determine its closed form representation in terms of the inverse regularized beta function.