In recent years the As-Rigid-As-Possible with Smooth Rotations (SR-ARAP [5]) technique has gained popularity in applications where an isometric-type of surface mapping is needed. The advantage of SR-ARAP is that quality of deformation results is comparable to more costly volumetric techniques operating on tetrahedral meshes. The SR-ARAP relies on local/global optimisation approach to minimise the non-linear least squares energy. The power of this technique resides on the local step. The local step estimates the local rotation of a small surface region, or cell, with respect of its neighbouring cells, so a local change in one cell’s rotation affect the neighbouring cell’s rotations and vice-versa. The main drawback of this technique is that the local step requires a global convergence of rotation changes. Currently the local step is solved in an iterative fashion, where the number of iterations needed to reach convergence can be prohibitively large and so, in practice, only a fixed number of iterations is possible. This trade-off is, in some sense, defeating the goal of SR-ARAP. We propose a linear-time closed-form solution for estimating the codependent rotations of the local step by solving a sparse linear system of equations. Our method is more efficient than state-of-the-art since no iterations are needed and optimised sparse linear solvers can be leveraged to solve this step in linear time. It is also more accurate since this is a closed-form solution. We apply our method to generate interactive surface deformation, we also show how a multiresolution optimisation can be applied to achieve real-time animation of large surfaces.