Lambert Projections

Figure 1. The Lambert Projection produces conformal maps with very low distortion of area, and hence produces maps with some of the lowest overall distortion parameters possible.

The Lambert projection (or, to be more precise, the Lambert Conformal Conic projection, but be advised that this complete name is rarely if ever used) is one of the most commonly used projections. As its full name implies, the Lambert projection is conformal, and thus it cannot be equivalent. However, it has just about the lowest distortion of area possible for a conformal projection, making it just about as close as you can get to a projection that is simultaneously equivalent and conformal. This combination of qualities makes the Lambert projection very attractive for many mapping tasks, and manifests itself in maps that make a great deal of intuitive sense. For example, the Lambert projection is the only commonly used conic projection that displays the poles as the points they truly are.

It should be noted that the Lambert projection is used in all zones in the State Plane Coordinate System (SPCS) that extend farther east-to-west than they do north-to-south. It is also used in most U.S. Geological Survey maps produced since 1957; even though many of the 7½ minutes quad maps produced since then indicate that they were created using a polyconic projection, in actual fact they were produced using a Lambert's (the USGS has been mislabeling 7½ minute maps for years; someday they'll get them all fixed).

Figure 2. Johann Heinrich Lambert (1728 - 1777).

The Lambert projection was developed by Johann Heinrich Lambert in 1772. Lambert was born in either Germany or France (depending on who you believe) in 1728 and died in Berlin in 1777 (that's his "official" portrait in Figure 2). Lambert was an extremely influential mathematician; his accomplishments are many and they still play an important role in contemporary, cutting edge mathematics (he developed the concepts of hyperbolic sines and cosines; he proved that pi was an irrational number; he did a great deal of work with a theory known as the parallel postulate, and this work proved to be the foundation for non-euclidean geometry, and so on). Lambert's accomplishments are even more impressive when you consider that he never received any formal schooling in mathematics; he was entirely self-taught.

• Form: Lambert projections are conic.

• Case: Lambert projections are secant.

Figure 3. The typical location of the lines of tangency in a Lambert projection.

• Aspect: Lambert projections have normal aspects. Given their secant case, the two lines of tangency in a Lambert projection fall along lines of latitude. These lines are typically placed about 1/6^th of the map's north-to-south extent south of the map's north edge, and an identical distance north of the map's south edge (Figure 3). This is a old cartographers rule-of-thumb (it dates back at least 100 years), and has become known as the one sixth rule or the rule of sixths.

• Variation Within Lambert Projections: Lambert projections differ from one another in their aspect. Usually, a cartographer will specify the projection's aspect by specifying the location of the two lines of latitude that serve as the projection's lines of tangency.

• Distortions
  • Shearing: The Lambert projection is conformal; the shapes of small areas are maintained much more accurately than they are in non-conformal projections.
  • Tearing: Lambert maps show lines of latitude as parallel curved lines and lines of longitude as straight lines radiating out from the pole (Figure 1). This gives the Lambert projection curved north and south edges (unless the projection is used to maps one of the poles, in which case the map has no edge over the pole -- the pole is shown as a single point) and straight east and west edges. Tearing occurs along these edges. Geodetic scientists sometimes recommend that Lambert projections not be used to cover areas larger than about 45 degrees of latitude and/or longitude, but given the overall low distortion characteristics of the Lambert projection, this recommendation is frequently ignored. Lambert projections are occasionally used to make interrupted maps, but this is fairly rare.
  • Compression: Lambert projections do distort areas; they are not equivalent. However, the amount of area distortion is minimal near the lines of tangency. Areas are artificially reduced between the lines of tangency and artificially expanded outside them.
  • Equivalence: Lambert projections do distort areas; they are not equivalent. However, the amount of area distortion is minimal near the lines of tangency. Areas are artificially reduced between the lines of tangency and artificially expanded outside them.
  • Conformality: The Lambert projection is conformal; the shapes of small areas are maintained much more accurately than they are in non-conformal projections.
  • Equidistance: Distances are correct along the lines of tangency. Between the lines of tangency, distances are artificially reduced; outside the lines of tangency, distances are artificially increased.
  • Azimuthality: Directions between points close together are relatively accurate because of the projection's conformality. However, the Lambert projection is not truly azimuthal; there is no point (or points) from which all directions are accurately depicted.
  
  
• Uses: The Lambert projection is very attractive because of its low overall distortion. It is very widely used. In fact, it is a standard used by the U.S. Geological Survey, and it is one of the fundamental projections used in the State Plane Coordinate System. Its lack of true azimuthality and equivalence are its greatest weaknesses; it is not a good choice for mapping projects where accurate directions and/or areas are vital. However, in mapping projects where shapes are critical, the Lambert projection's conformality and low area distortion make it a very attractive option.