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\textit{Ab initio} calculations of electron-phonon interactions including thepolar Fr\"ohlich coupling have advanced considerably in recent years. TheFr\"ohlich electron-phonon matrix element is by now well understood in the caseof bulk three-dimensional (3D) materials. In the case of two-dimensional (2D)materials, the standard procedure to include Fr\"ohlich coupling is to employCoulomb truncation, so as to eliminate artificial interactions between periodicimages of the 2D layer. While these techniques are well established, thetransition of the Fr\"ohlich coupling from three to two dimensions has not beeninvestigated. Furthermore, it remains unclear what error one makes whendescribing 2D systems using the standard bulk formalism in a periodic supercellgeometry. Here, we generalize previous work on the \textit{ab initio}Fr\"ohlich electron-phonon matrix element in bulk materials by investigatingthe electrostatic potential of atomic dipoles in a periodic supercellconsisting of a 2D material and a continuum dielectric slab. We obtain aunified expression for the matrix element, which reduces to the existingformulas for 3D and 2D systems when the interlayer separation tends to zero orinfinity, respectively. This new expression enables an accurate description ofthe Fr\"ohlich matrix element in 2D systems without resorting to Coulombtruncation. We validate our approach by direct \textit{ab initio}density-functional perturbation theory calculations for monolayer BN andMoS$_2$, and we provide a simple expression for the 2D Fr\"ohlich matrixelement that can be used in model Hamiltonian approaches. The formalismoutlined in this work may find applications in calculations of polarons,quasiparticle renormalization, transport coefficients, and superconductivity,in 2D and quasi-2D materials.

Unified ab initio description of Fr\"ohlich electron-phonon interactions in two-dimensional and three-dimensional materials