The most common scheme for estimating the orientation field of space-varying directional textures is based on a local nonlinear spatial averaging of the gradient field. This leads to locally biased orientations, especially in regions of nonlinearly distributed convergence, asymmetrically distributed curvature, or geometry superposition. In this paper, we propose an orientation estimation framework that is invariant toward the local geometry. Instead of applying the spatial averaging in the initial space, it is performed in a flattened space which is obtained from a parametric space transformation model deduced from the reconstruction of local hypersurfaces. The flattening can either be applied to the initial gradient field or to the intensity image, on which the flattened gradient field is computed. The process is iterated in order to define more accurate space transformations leading to refined orientation estimation. The computational efficiency is improved without noticeable loss of orientation accuracy by a patch-based approach. Experiments, both on synthetic as well as real-life fingerprint and fibrous material images, exhibit enhanced performance in comparison with conventional methods such as the structure tensor method.