Do equal-area projections really preserve areas?

Abstract

Map projections can be developed to preserve areas, angles, and lengths. Although the lengths are preserved in a certain direction, and the angle preserving is valid for differential quantities, the equal-area property is valid for both the differential and the finite quantities. However, areas of finite shapes are not exact in the projection plane except for regions bounded by meridians and parallels. The reason for this is that the great circle arcs that constitute the shape of the sphere and the line segments in the plane are not coincident. This effect is significant when a certain length of great circles that belong to a shape is exceeded. In this article, using variable-sized equilateral spherical triangles defined around a point, area errors are analyzed for a set of points defined at regular intervals for four selected projections. The area error varies depending on the size and position of the spherical triangle. For the selected projections, it has been shown at which intervals of point densification the area errors can be reduced. It was seen that the most sensitive projection in terms of area error was the equal-area cylindrical projection.