Gall’s Stereographic projects the earth onto a cylinder. Since the point of projection is not at the earth’s center, the great exaggeration of the extreme latitudes is avoided. The result is a pleasing picture of the earth with area and shape distortions reduced. The projection ‘averages’ the area distortions of conformal projections with the shape distortions of equal-area maps. This ‘compromising’ result pleases the eye but has no other strict cartographic virtue.
Van der Grinten projects the entire earth onto a circle. It assigns the greatest distortion to the areas very near the poles, leaving the rest of the earth to be viewed with minimized distortions. Usually the polar areas are cut off the map.