The “Thimm trick” of symplectic geometry extends a Hamiltonian K-action on a symplectic manifold to a K × T-action – except at some bad points, where the action is not well-defined, but “its moment map” still is. (The map isn’t differentiable, just continuous, so doesn’t define a Hamiltonian vector field.) The algebraic geometers have an analogous construction called the “Vinberg asymptotic cone.” I will show (1) that there in the complex algebraic case, there is a natural continuous map from the original space to its asymptotic cone and (2) in a certain limit the Thimm moment map is the composite of this with the honest moment map on the asymptotic cone. Even when these maps are smooth, they can be interesting, in showing distinct projective varieties are symplectomorphic.