Many functions can map images to the sphere for use as environment maps or spherical panoramas. We develop a new metric that asymptotically measures how well these maps use a given number of samples to provide the greatest worst-case frequency content of the image everywhere over the sphere. Since this metric assumes perfect reconstruction filtering even with highly anisotropic maps, we define another, conservative measure of sampling efficiency that penalizes anisotropy using the larger singular value of the mapping’s Jacobian. With these metrics, we compare spherical maps used previously in computer graphics as well as other mappings from cartography, and propose several new, simple mapping functions (dual equidistant and polar-capped maps) that are significantly more efficient and exhibit less anisotropy. This is true with respect to either efficiency metric, which we show agree in the worst case for all but one of the spherical maps presented. Although we apply the metrics to spherical mapping they are useful for analyzing texture maps onto any 3D surface.