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A full-Stokes model provides the most accurate but also the most expensive representation of ice sheet dynamics. The Blatter-Pattyn model is a widely used less expensive approximation that is valid for ice sheets characterized by a small aspect ratio. Here we introduce a novel transformation of the Stokes equations into a form that closely resembles the Blatter-Pattyn equations. The transformed exact Stokes equations only differ from the approximate Blatter-Pattyn equations by a few additional terms, while their variational formulations differ only by the presence of a single term in each horizontal direction (one term in 2D and two terms in 3D). Specifically, the variational formulations differ only by the absence (or the neglect) of the vertical velocity in the second invariant of the strain rate tensor in the Blatter-Pattyn model when compared to the Stokes case. Here we make use of the new transformation in two different ways. First, we consider incorporating the transformed equations into a code that can be very easily converted from a Stokes to a Blatter-Pattyn model, and vice-versa, simply by switching these terms on or off. This may be generalized so that the Stokes model is switched on adaptively only where the Blatter-Pattyn model loses accuracy, hopefully retaining most of the accuracy of the Stokes model but at a lower cost. Second, the key role played by the vertical velocity in converting the transformed Stokes model into the Blatter-Pattyn model motivates new approximations that improve on the Blatter-Pattyn model, heretofore the best approximate ice sheet model. These applications require the use of a grid that enables the discrete continuity equation to be invertible for the vertical velocity in terms of the horizontal velocity components. Examples of such grids, such as the first-order P1-E0 grid and the second-order P2-E1 grid are given in both 2D and 3D. It should be noted, however, that the transformed Stokes model has the same type of gravity forcing as the Blatter-Pattyn model, i.e., determined by the slope of the ice sheet's upper surface, thereby forgoing some of the grid-generality of the traditional formulation of the Stokes model. This is not a serious disadvantage, however, since in practice it has not impaired the widespread use of the Blatter-Pattyn model.