## Description

The Lambert azimuthal equal-area projection maintains land features at their true relative sizes while simultaneously maintaining a true sense of direction from the center. The world is projected onto a flat surface from any point on the globe. Although all aspects are possible (equatorial, polar, and oblique), the one used most commonly is the polar aspect. This projection is best suited for individual land masses that are symmetrically proportioned, either round or square.

The Lambert azimuthal equal-area projection was developed by Johann H. Lambert in 1772. It is available in ArcGIS Pro 1.0 and later and in ArcGIS Desktop 8.0 and later.

## Projection properties

The subsections below describe the Lambert azimuthal equal-area projection properties.

### Graticule

Lambert azimuthal equal-area is an azimuthal projection with three aspects: polar, equatorial, and oblique.

In the polar aspect, the meridians project as straight lines originating at the pole and the angles between them are true. The parallels are shown as unequally spaced concentric circular arcs and their spacing decreases with the distance from the center. All graticule line intersections are 90 degrees. The opposite pole is projected as a circle and presents the edge of the map. The graticule is symmetric across any meridian.

In the equatorial aspect, the equator and the central meridian are projected as two perpendicular straight lines. Two meridians, 90 degrees east and west of the central meridian, project as a circle. Other meridians are complex curves. Their spacing decreases away from the central meridian. All parallels are complex curves concave toward the nearest pole and unequally spaced along the central meridian. The spacing of the parallels decreases away from the equator. Both poles project as points. The antipodal point of the projection's center projects as a circle and presents the edge of the map. The graticule is symmetric across the equator and the central meridian.

In the oblique case, only the central meridian and antimeridian project as straight lines. The other meridians and parallels are complex curves. The meridians intersect at poles and are projected as a point. The parallels are unequally spaced along the central meridian and their spacing decreases away from the projection's center. The graticule is symmetric across the central meridian.

### Distortion

Lambert azimuthal equal-area is an equal-area (equivalent) projection. Shapes, directions, angles, and distances are generally distorted. Scale and directions are true only at the center of the projection. The scale decreases with the distance from the center along the radii and increases from the center perpendicularly to the radii, resulting in small shapes compressed radially from the center and elongated perpendicularly. The general pattern of distortion is radial.

## Usage

Although the Lambert azimuthal equal-area projection can display the entire globe, its practical usage is typically limited to a hemisphere. This projection is most commonly used for thematic mapping of polar regions. The oblique aspect is sometimes used for individual land masses that are symmetrically proportioned, either round or square.

## Variants

There are two variants available in ArcGIS.

- Lambert azimuthal equal-area is available in ArcGIS Pro 1.0 and later and in ArcGIS Desktop 8.0 and later.
- Lambert azimuthal equal-area auxiliary sphere is available in ArcGIS Pro 1.0 and later and in ArcGIS Desktop 9.3 and later. This variant does not support the ellipsoid and uses sphere-based equations with a sphere specified by the Auxiliary Sphere Type parameter. Although resulting coordinates will not match the true ellipsoidal solution, the Lambert azimuthal equal-area auxiliary sphere variant maintains the equal-area property when the Auxiliary Sphere Type 3 is used. Otherwise, areas are not maintained for ellipsoids using this variant.

## Limitations

Only the Lambert azimuthal equal-area variant supports ellipsoids.

## Parameters

Lambert azimuthal equal-area parameters are as follows:

- False Easting
- False Northing
- Central Meridian
- Latitude Of Origin

Lambert azimuthal equal-area auxiliary sphere parameters are as follows:

- False Easting
- False Northing
- Central Meridian
- Latitude Of Origin
- Auxiliary Sphere Type, with values as follows:
- 0 = Use semimajor axis of the geographic coordinate system
- 1 = Use semiminor axis
- 2 = Calculate and use authalic radius
- 3 = Use authalic radius and convert geodetic latitudes to authalic latitudes

##### Note:

If the geographic coordinate system uses a sphere, the Auxiliary Sphere Type uses the radius of the sphere in all four cases.

## Sources

Snyder, J. P. (1987). Map Projections: A Working Manual. U.S. Geological Survey Professional Paper 1395. Washington, DC: United States Government Printing Office.

Snyder, J. P. (1993). Flattening the Earth. Two Thousand Years of Map Projections. Chicago and London: University of Chicago Press.

Snyder, J. P. and Voxland, P. M. (1989). An Album of Map Projections. U.S. Geological Survey Professional Paper 1453.Washington, DC: United States Government Printing Office.