  Consider being given a mapping \phi from the unit sphere S^{d-1}, d>2, to the
smooth boundary of a simply-connected region \Omega in R^d. We consider the
problem of constructing an extension \Phi from the unit ball B_d to \Omega. The
mapping is required to be 1-1 and continuously differentiable with a
nonsingular Jacobian matrix. We discuss ways of obtaining initial guesses for
such a mapping \Phi and of then improving it by an iteration method.
