Lambert Azimuthal Equal-Area Projection

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Introduction

Lambert Azimuthal Equal-Area Projection is an equal-area azimuthal projection proposed by German mathematician Johann Heinrich Lambert in 1772. This projection maps points on the earth’s surface onto a plane through spherical projection, keeping the projection area unchanged, that is, the area ratio of the area before and after the projection is strictly equal. There is no deformation at the center of the projection, and as the distance from the center increases, the angle and shape deformation gradually increase, but the area deformation is always zero.

Projection Composition

**Geometric construction: **This projection takes the projection center point as the reference, regards the earth’s surface as a sphere, and maps the points on the sphere to the plane through spherical projection. The projection center point has no deformation. As the distance from the center point increases, the angle and shape deformation gradually increase, but the area deformation is always zero.
**Mathematical model: **The Lambert azimuthal projection constructs a mathematical model based on the equal area characteristic to ensure that the area ratio of the area before and after the projection is strictly equal. This characteristic makes this projection have significant advantages in map applications that need to accurately reflect the area ratio.
Projection characteristics:
- Equal area: The area ratio of the area before and after the projection is strictly equal, which is suitable for map applications that need to accurately reflect the area ratio.
- Deformation law: As the distance from the projection center point increases, the shape and angle deformation gradually increase. In polar regions, the deformation is small, which can better maintain the shape and area characteristics of polar regions.
**Application orientation: **The original intention of the design of this projection is to meet the needs of polar region mapping, great circle route analysis, and regional area comparison. In these scenarios, the area-preserving property makes the Lambert azimuthal projection an ideal choice.

Pros

Equal area characteristics: The area ratio of the area before and after projection is strictly equal, which is suitable for map applications that need to accurately reflect the area ratio, such as resource distribution maps, population density maps, etc.
Excellent performance in polar regions: In polar regions, the Lambert azimuthal projection has less deformation and can better maintain the shape and area characteristics of polar regions, which is suitable for mapping and analysis in polar regions.
Great circle routes are intuitive: Great circle routes appear as straight lines on the projection plane, which is convenient for analyzing and planning great circle routes and improving the efficiency of navigation and path planning.

Cons

Shape and angle distortion: Shape and angle distortion increases with distance from the projection center. In areas far from the projection center, the shape and angle of the map may differ greatly from the actual earth’s surface.
Limited applicability: The Lambert azimuthal projection is best suited for areas centered on the projection center point and may not be the best choice for maps that span a large area or contain multiple different areas.
Directional changes: The direction of all points except the projection center point changes, which may affect some map applications that require accurate directional information.

Application Scenario

Lambert Azimuthal Equal-Area Projection is suitable for high-precision mapping of polar regions to maintain the shape and area characteristics of the polar regions, great circle route analysis and path planning in the field of aviation and navigation, resource distribution and ecological protection, and other scenarios that require accurate comparison of regional areas, as well as professional analysis in geological and geophysical research that requires no deformation of the area.

Example

Lambert Azimuthal Equal-Area Projection.

The Lambert azimuthal equal-area projection with Tissot’s indicatrix of deformation.

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