Image-based approximation of derivatives of traditional differential metrics of angular distortion in map projections

Map projections are imaging procedures used to depict geographic features. We adopt the traditional differential metric and exploit the intrinsic image properties of map projections to establish an image-based differential metric for evaluating distortions in map projections, obtaining an effective, practical, and relatively accurate metric. We use bivariate polynomial functions to approximate the forward and inverse formulae of map projections. Thereafter, the proposed metric is conveniently calculated using the partial derivatives of the approximate forward functions based on polynomial functions, while complicated differential calculations are avoided. Moreover, multiple sampling and image filters mitigate the influence of imaging noise and achieve a high computation precision. Experiments were conducted using the NASA G.Projector mapping software to generate images from more than 200 map projections. Explicit equations of map projections were not required owing to the use of the mapping software. These images were then evaluated using the proposed metric through an implementation in the Julia programming language. The corresponding results confirmed that the proposed metric avoided the drawbacks of the great circle arc metric and provided considerably low errors (1.12° on average) and high consistency (0.999 on average) with respect to the traditional differential metric. Although there were errors, experimental results indicated that feasibility and high usability were achieved by the image-based method for evaluating distortions in small-scale map projections.