Lambert conformal conic projection
A Lambert conformal conic projection (LCC) is a conic map projection, which is often used for aeronautical charts. In essence, the projection superimposes a cone over the sphere of the Earth, with two reference parallels secant to the globe and intersecting it. This minimizes distortion from projecting a three dimensional surface to a two-dimensional surface. There is no distortion along the standard parallels, but distortion increases further from the chosen parallels. As the name indicates, maps using this projection are conformal.
Pilots favor these charts because a straight line drawn on a Lambert conformal conic projection approximates a great-circle route between endpoints. The European Environment Agency recommends its usage for conformal pan-European mapping at scales smaller or equal to 1:500,000[1].
In the United States, the National Geodetic Survey's "State Plane Coordinate System of 1983" uses Lambert Conformal Conic Projection to define the grid-coordinate systems used in several States (primarily those that are elongated west to east, like Tennessee). The Lambert projection is relatively easy to use: conversions from Geodetic (latitude/longitude) to State Plane Grid coordinates involve trigonometric equations that are fairly straightforward and which can be solved on most scientific calculators, especially programmable models (see NGS-NOS-5.PDF in the External Links section of this article for details). Lambert projection as used in CCS83 produces maps in which scale errors produced by the inherent distortion are limited to 1 part in 10,000.
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[edit] History
The Lambert conformal conic is one of several map projection systems developed by Johann Heinrich Lambert, an 18th century Swiss mathematician, physicist, philosopher, and astronomer.
[edit] Transformation
Coordinates from a spherical datum can be transformed into Lambert conformal conic projection coordinates with the following formulas[2], where λ is the longitude, λ0 the reference longitude, φ the latitude, φ0 the reference latitude, and φ1 and φ2 the standard parallels:
- x = ρsin[n(λ − λ0)]
- y = ρ0 − ρcos[n(λ − λ0)]
where
![n = \frac{\ln(\cos \phi_1 \sec \phi_2)}{\ln [\tan (\frac14 \pi + \frac12 \phi_2) \cot (\frac14 \pi + \frac12\phi_1)]}](http://upload.wikimedia.org/math/2/7/9/279721ee5afba31bcf501e61e2d96b2c.png)
Formulæ for ellipsoidal datums are more involved.
[edit] See also
- Albers projection
- Lambert cylindrical equal-area projection
- Lambert azimuthal equal-area projection
- Johann Heinrich Lambert
[edit] References
- ^ "Short Proceedings of the 1st European Workshop on Reference Grids, Ispra, 27-29 October 2003". European Environment Agency. 2004-06-14. p. 6. http://eusoils.jrc.ec.europa.eu/Projects/Alpsis/Docs/ref_grid_sh_proc_draft.pdf. Retrieved 2009-08-27.
- ^ Weisstein, Eric. "Lambert Conformal Conic Projection". Wolfram MathWorld. Wolfram Research. http://mathworld.wolfram.com/LambertConformalConicProjection.html. Retrieved 2009-02-07.
[edit] External links
- Table of examples and properties of all common projections, from radicalcartography.net
- An interactive Java Applet to study the metric deformations of the Lambert Conformal Conic Projection
- This document from the U.S. National Geodetic Survey describes the State Plane Coordinate System of 1983, including details on the equations used to perform the Lambert Conformal Conic and Mercator map projections of CCS83
- Lambert Conformal Conic to Geographic Transformation Formulae from Land Information New Zealand
