Lambert conformal conic projection

Pilots use aeronautical charts based on LCC because a straight line drawn on a Lambert conformal conic projection approximates a great-circle route between endpoints for typical flight distances. The US systems of VFR (visual flight rules) sectional charts and terminal area charts are drafted on the LCC with standard parallels at 33°N and 45°N.

The European Environment Agency[3] and the INSPIRE specification for coordinate systems [4] recommends using this projection (also named ETRS89-LCC) for conformal pan-European mapping at scales smaller or equal to 1:500,000. In Metropolitan France, the official projection is Lambert-93,[5] a Lambert conic projection using RGF93 geodetic system[6] and defined by references parallels that are 44°N and 49°N.

The National Spatial Framework for India uses Datum WGS84 with a LCC projection and is a recommended NNRMS standard. Each state has its own set of reference parameters given in the standard.[7]

The U.S. National Geodetic Survey's "State Plane Coordinate System of 1983" uses the Lambert conformal conic projection to define the grid-coordinate systems used in several states, primarily those that are elongated west to east such as 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.[8] The projection as used in CCS83 yields maps in which scale errors are limited to 1 part in 10,000.

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.[1]

Coordinates from a spherical datum can be transformed into Lambert conformal conic projection coordinates with the following formulas,[9] where λ is the longitude, λ₀ the reference longitude, φ the latitude, φ₀ the reference latitude, R the radius of the Earth and φ₁ and φ₂ the standard parallels:

${\displaystyle {\begin{aligned}x&=\rho \sin \left[n\left(\lambda -\lambda _{0}\right)\right]\\y&=\rho _{0}-\rho \cos \left[n\left(\lambda -\lambda _{0}\right)\right]\end{aligned}}}$

where

${\displaystyle {\begin{aligned}n&={\frac {\ln \left(\cos \varphi _{1}\sec \varphi _{2}\right)}{\ln \left[\tan \left({\frac {1}{4}}\pi +{\frac {1}{2}}\varphi _{2}\right)\cot \left({\frac {1}{4}}\pi +{\frac {1}{2}}\varphi _{1}\right)\right]}}\\\rho &=RF\cot ^{n}\left({\tfrac {1}{4}}\pi +{\tfrac {1}{2}}\varphi \right)\\\rho _{0}&=RF\cot ^{n}\left({\tfrac {1}{4}}\pi +{\tfrac {1}{2}}\varphi _{0}\right)\\F&={\frac {\cos \varphi _{1}\tan ^{n}\left({\frac {1}{4}}\pi +{\frac {1}{2}}\varphi _{1}\right)}{n}}\end{aligned}}}$

If one standard parallel is used (i.e. ${\displaystyle \varphi _{1}=\varphi _{2}}$), the formula for n above is indeterminate, but then ${\displaystyle n=\sin(\varphi _{1})}$.[10]

Formulae for ellipsoidal datums are more involved.[10]

• List of map projections
• Albers projection
• Lambert cylindrical equal-area projection
• Lambert azimuthal equal-area projection
• Spatial reference system

Wikimedia Commons has media related to Lambert conformal conic projection.

• 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