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Description: The orientations of planar rock layers are fundamental to our understanding of structural geology and stratigraphy. Remote-sensing platforms including satellites, unmanned aerial vehicles (UAVs), and LIDAR scanners are increasingly used to build three-dimensional models of structural features on Earth and other planets. Remotely-gathered orientation measurements are straightforward to calculate but are subject to uncertainty inherited from input data, differences in viewing geometry, and the regression process, complicating geological interpretation. Here, we improve upon the present state of the art by developing a generalized means for computing and reporting errors in strike-dip measurements from remotely sensed data. We outline a general framework for representing the error space of uncertain orientations in Cartesian and spherical coordinates and develop a principal-component analysis (PCA) regression method which captures statistical errors independent of viewing geometry and input data structure. We also build graphical techniques to visualize the uniqueness and quality of orientation measurements, and a process to increase statistical power by jointly fitting bedding planes under the assumption of parallel stratigraphy. These new techniques are validated by comparison of field-gathered orientations with minimally-processed satellite imagery of the San Rafael Swell, Utah and UAV imagery from the Naukluft Mountains, Namibia. We provide software packages supporting planar fitting and the visualization of error distributions. This method provides a means to increase the precision and comparability of structural measurements gathered using a new generation of remote-sensing techniques.