Conic Family·Inventor: Johann Heinrich Lambert·Year: 1772

Lambert Conformal Conic (LCC)

"Conformal conic projection standard for mid-latitude aeronautical navigation charts"

Lambert’s mathematical tour de force projects the sphere onto a cone intersecting the globe at two standard parallels. Meridians are straight lines converging at the pole; parallels are concentric circular arcs.

Forward Mathematical Formulation

X-COORDINATE:

x = \rho \sin[n(\lambda - \lambda_0)]

Y-COORDINATE:

y = \rho_0 - \rho \cos[n(\lambda - \lambda_0)]

DIFFERENTIAL METRIC:

\rho = F \left[ \tan(\pi/4 + \phi/2) \right]^{-n}, \quad n = \frac{\ln(\cos \phi_1 \sec \phi_2)}{\ln[\tan(\pi/4+\phi_2/2)\cot(\pi/4+\phi_1/2)]}

Cartographic Application & Critique

Primary Applications: FAA Aeronautical Sectional Charts, State Plane Coordinate System (SPCS east-west zones), USGS topographic maps.

Critical Analysis: The undisputed king of mid-latitude continental aviation: a straight line drawn on a Lambert Conformal Conic chart approximates a true great-circle flight path.

TISSOT INDICATRIX (Lat: 60°N)Area: 1.02×

MERIDIAN (h)

1.13

PARALLEL (k)

0.90

ANGULAR (2θ')

13.2°

Tissot Profile Across Latitudes

LATITUDE	SCALE h	SCALE k	AREA hk	2θ' DEFORM
33°N	1.00	1.00	1.00×	0.0°
45°N	1.00	1.00	1.00×	0.0°
39°N	0.98	0.98	0.97×	0.0°
60°N	1.08	1.08	1.17×	0.0°

Interdisciplinary Geometric Links

JuralogiumREGULATES

National Airspace Boundaries

LunologiumTRACES

Great Circle Great Geodesics

Academic Provenance: Lambert, J. H. (1772)