The present invention discloses methods for lossless shearing and/or rotation of two-dimensional (2D) data, including digital images, with minute discrete angular increments, carried out only by permutations in the Fourier frequency domain, by exploiting the natural shear occurring as a result of computing a single one-dimensional discrete Fourier transform (DFT) of 2D arrays. Rotations in general, especially for oblong arrays, occur on elliptical paths. Circular rotation, by an angle of arctan(1/width), is achieved on square arrays. When each dimension is multiple of a smaller N, the rotation/shear angle can be increased to arctan(2N/width). Rotation steps can be repeated in long, animation-like series, with neither loss nor degradation of the Fourier content; so much so that tracing the steps back does restore the original data with remarkable precision.