Title: Flexion and Skewness in Map Projections of the Earth

Abstract: Tissot indicatrices have provided visual measures of local area and isotropy distortions. Here we show how large scale distortions of flexion (bending) and skewness (lopsidedness) can be measured. Area and isotropy distortions depend on the map projection metric, flexion and skewness, which manifest themselves on continental scales, depend on the first derivatives of the metric. We introduce new indicatrices that show not only area and isotropy distortions but flexion and skewness as well. We present a table showing error measures for area, isotropy, flexion, skewness, distances, and boundary cuts allowing us to compare different world map projections. We find that the Winkel-Tripel projection (already adopted for world maps by the National Geographic), has low distortion on most measures and excellent quality overall.