Solution of Laplace equation on non axially symmetrical volumes

The homogeneity of the magnetic field plays a fundamental role in MRI. Standard shimming techniques of the magnetic field are usually applied on volumes such as spheres or (less frequently) on surfaces of revolution (oblate and prolate spheroids) and are based on well-known solutions of the Laplace equation. We present a complete mathematical formalism for the solution of the Laplace equation with Dirichlet conditions defined on a tri-axial ellipsoid through the transformation of the equation in ellipsoidal coordinates. The importance of the ellipsoid lies in the fact that this surface can be more easily conformed to most districts of the human body (e.g. extremities) and this is of interest for dedicated MRI systems.