Lambert Azimuthal Equal-area Projection - Applications

Applications

The Lambert azimuthal projection was originally conceived as an equal-area map projection. It is now also used in disciplines such as geology to plot directional data, as follows.

A direction in three-dimensional space corresponds to a line through the origin. The set of all such lines is itself a space, called the real projective plane in mathematics. Every line through the origin intersects the unit sphere in exactly two points, one of which is on the lower hemisphere . (Horizontal lines intersect the equator in two antipodal points. It is understood that antipodal points on the equator represent a single line. See quotient topology.) Hence the directions in three-dimensional space correspond (almost perfectly) to points on the lower hemisphere. The hemisphere can then be plotted as a disk of radius using the Lambert azimuthal projection.

Thus the Lambert azimuthal projection lets us plot directions as points in a disk. Due to the equal-area property of the projection, one can integrate over regions of the real projective plane (the space of directions) by integrating over the corresponding regions on the disk. This is useful for statistical analysis of directional data.

Not only lines but also planes through the origin can be plotted with the Lambert azimuthal projection. A plane intersects the hemisphere in a circular arc, called the trace of the plane, which projects down to a curve (typically non-circular) in the disk. One can plot this curve, or one can alternatively replace the plane with the line perpendicular to it, called the pole, and plot that line instead. When many planes are being plotted together, plotting poles instead of traces produces a less cluttered plot.

Researchers in structural geology use the Lambert azimuthal projection to plot crystallographic axes and faces, lineation and foliation in rocks, slickensides in faults, and other linear and planar features. In this context the projection is called the equal-area hemispherical projection. There is also an equal-angle hemispherical projection defined by stereographic projection.

The discussion here has emphasized the lower hemisphere, but some disciplines prefer the upper hemisphere . Indeed, any hemisphere can be used to record the lines through the origin in three-dimensional space.