in [@Mertens2012] $$\label{eq:phi}
p=1-\exp(-nV_p),$$ where $V_p$ is the volume or area of each individual particle for the 3D or 2D problems, respectively.

Percolation problems for elongated particles have been extensively studied both in 3D and 2D geometries. Investigated 3D geometries include randomly oriented overlapping ellipsoids [@Garboczi1995; @Yi2004], overlapping capped cylinders [@Balberg1984; @Balberg1984a; @Balberg1987; @DeBondt1992], soft-core capped cylinders [@Berhan2007] and soft-core prisms [@Saar2002]. For 2D percolation, randomly oriented overlapping ellipses [@Xia1988; @Yi2002], and rectangles [@Li2013] have also been considered. Analytical approximations and Monte Carlo (MC) simulations have been applied to derive the percolation thresholds for overlapping ellipsoids [@Yi2004] and overlapping ellipses [@Yi2002].

The most popular models for describing percolation problems have been the excluded volume (3D) or excluded area (2D) approaches. It has been conjectured that the number density at the geometrical percolation threshold, $n_c$, is inversely proportional to the excluded volume (3D) or excluded area (2D) of the particles, $V_{ex}$, [@Balberg1984a] $$\label{eq:Inverse}
n_c \propto V^{-1}_{ex}.$$

The value of $V_{ex}$ depends upon the system dimensionality, the geometry of the particles and their relative orientations.

In 3D systems for oriented cylinders of length $l$ and diameter $d$, the excluded volume $V_{ex}$ is given by $$\label{eq:Vex3D}
V_{ex}=2d l^2\left[\langle \sin\theta \rangle+\pi/a+(2\pi/3)/a^2\right],$$ where $\theta$ is 