Laplace eigenvalues on regular polygons: A series in 1/N

For regular polygons Pry inscribed in a circle, the eigenvalues of the Laplacian converge as N -> infinity to the known eigenvalues on a circle. We compute the leading terms of lambda(N)/lambda in a series in powers of 1/N, by applying the calculus of moving surfaces to a piecewise smooth evolution from the circle to the polygon. The 0 (1/N-2) term comes from Hadamard's formula, and reflects the change in area. This term disappears if we "transcribe" the polygon, scaling it to have the same area as the circle. (C) 2011 Published by Elsevier Inc.