Percolation properties of random ellipses

We compute the percolation concentration for identical ellipses with aspect ratio b/a in a two-dimensional plane where both the centers and orientations are random. We have obtained a comprehensive set of results ranging from circles to needles which can be well fit by the interpolation formula pc=(1+4y)/(19+4y), where y=b/a+a/b. We also obtain results for random centers and random orientations along the two principal directions which are virtually indistinguishable from the previous case. These results are used to critique the various effective-medium theories that have been developed for the electrical conductance and elastic moduli of sheets containing random elliptical inclusions. An interpolation formula is developed that appears superior to all these effective-medium theories.