MHD equilibria all the way to the separatrix

It is typical for one to define the boundary of a 3D MHD equilibrium calculation to be a flux surface a short distance within the separatrix rather than the separatrix itself. This is because current software such as DESC or VMEC relies upon smooth Fourier representations for the boundary which preclude an accurate and cheap representation of the sharp edges at the x-points of the separatrix. Regardless, researchers may desire an equilibrium calculation that extends all the way to the separatrix, particularly for situations such as stellarators with an island divertor. In order to accomplish this, we present a novel method for mathematically representing a smoothly nested volume enclosed by a surface with sharp edges. Whereas DESC completely maps the interior of a unit disc to smooth nested tori using Fourier-Zernike bases, we present a similar map to smooth nested surfaces except for an outermost surface with sharp edges by incorporating an intermediate analytic map from the unit disc to a regular N-gon using a hypergeometric function. We demonstrate that this method represents typical stellarator separatrices such as that of W7-X much more accurately and cheaply than conventional Fourier representations.