Wrap a cone around the Earth so that it intersects along two standard parallels, then project geodetic points onto the developed plane — the LCC projection. Adjust the parallels, central meridian and reference latitude to see the cone constant n, planar (x, y) coordinates, scale factor and distortion update in real time. This is the mid-latitude workhorse used by US SPCS, ICAO ONC and operational weather charts.
Parameters
Standard parallel 1 φ₁
°
Lower-latitude parallel where the cone meets the globe
Standard parallel 2 φ₂
°
Higher-latitude parallel. Setting φ₁=φ₂ gives a tangent cone
Central meridian λ₀
°
Central longitude of the projection. SPCS picks a value near the state centre
Reference latitude φ₀
°
Latitude that defines the false-northing origin on the plane
Point latitude φ
°
Latitude of the geodetic point to project onto the plane
Point longitude λ
°
Longitude of the geodetic point to project onto the plane
Earth radius R
km
Spherical Earth radius. WGS84 mean radius is 6371 km
Results
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Cone constant n
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Plane X (km)
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Plane Y (km)
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Scale factor k
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Distortion (%)
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Cone-apex distance (km)
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Cone projection geometry
The globe intersected by a cone that meets the surface along the two standard parallels (φ₁, φ₂). The projection point (red) is mapped to the cone, and unrolling the cone gives the planar (x, y) coordinates plotted below.
Scale factor vs latitude (distortion curve)
Meridian / parallel grid in the projected plane
Theory & Key Formulas
$$n = \frac{\ln(\cos\phi_1/\cos\phi_2)}{\ln(\tan(\pi/4+\phi_2/2)/\tan(\pi/4+\phi_1/2))},\quad x = \rho\sin(n(\lambda-\lambda_0))$$
n is the cone constant (0<n<1), ρ the radius from the cone apex, (x, y) the plane coordinates (km), (φ, λ) the geodetic coordinates, λ₀ the central meridian, φ₁ and φ₂ the standard parallels.
$$\rho = R\,F/\tan^{n}(\pi/4+\phi/2),\qquad F = \cos\phi_1\,\tan^{n}(\pi/4+\phi_1/2)/n$$
ρ is the in-plane distance from the cone apex to latitude φ. F is the normalisation that forces k=1 along φ₁. R is the Earth radius.
k is the scale factor along the parallel. k<1 between the standard parallels, k>1 outside, k=1 along φ₁ and φ₂. Because the projection is conformal, the meridian scale equals k at the same point.
Lambert Conformal Conic — The Mid-Latitude Standard for Aviation and Weather Maps
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Lambert Conformal Conic is one of those map projections, right? How does it differ from Mercator or UTM?
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Think of it as "throwing a cone over the area you care about." Mercator uses a cylinder tangent to the equator and UTM uses a series of transverse cylinders, but Lambert uses a cone that cuts the Earth along two parallels (φ₁, φ₂) in the mid-latitudes. It is the standard projection for the US State Plane Coordinate System, ICAO ONC charts and operational weather charts. Leave the sliders at φ₁=33°, φ₂=45° and n comes out around 0.6305 — that number describes how widely the cone opens.
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So n=0.6305 means the cone opens up to roughly 63% of a full circle?
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Exactly — when you unroll the cone into a plane, the 360° of longitude maps to an arc of n×360° ≈ 227°. The limits make this intuitive. Push φ₁ towards 0 and n goes to 0: the cone becomes almost a cylinder, the Mercator limit. Push φ₁ and φ₂ both close to 89° and n approaches 1: the cone collapses to a near-plane, the polar stereographic limit. LCC is the conformal interpolation between these two extremes, optimised for mid latitudes.
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"Conformal" means it preserves angles — but the tool still shows some distortion?
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Good catch. "Conformal" means that around any point a tiny circle is mapped to a tiny circle, never to an ellipse. The local angles are preserved. But the size of that little circle changes with latitude, and the scale factor k captures that. Between the two standard parallels (φ₁, φ₂) you get k<1 (shrinkage); outside, k>1 (expansion). On the distortion curve below, at φ=40° you should see k ≈ 0.994, which is about 0.5% shrinkage.
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So bracketing the area with two standard parallels really does cut distortion. How does SPCS pick them?
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Each state sets φ₁ and φ₂ about 1/6 of the way in from the southern and northern edges of the zone. Colorado central (roughly 37°-41° N) uses φ₁=37°50′ and φ₂=39°45′, for example. That keeps k within 0.9999-1.0001 across the entire state — distance error on the order of 1 part in 10,000. A 1:24,000 topographic sheet will not be off by even 10 cm. ICAO ONC sheets do the same: the standard parallels sit inside the sheet edges so that |k-1| stays under 0.05%.
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If I set φ₁=φ₂ I get a "tangent cone" — how is that different from a "secant cone"?
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A tangent cone (φ₁=φ₂) has just one standard parallel where k=1, and distortion grows away from that single line. A secant cone (φ₁≠φ₂, the case this tool computes by default) has two such lines — between them k<1, outside them k>1. You get to spread the distortion symmetrically from both edges towards the middle. France's old NTF used a tangent Lambert (φ₀=48°); today's RGF93 Lambert-93 uses a secant cone with φ₁=44°, φ₂=49°. In practice almost every modern LCC system uses the secant form.
Frequently Asked Questions
Mid-latitude regions (roughly 30°-60° N or S) that extend east-west. The US State Plane Coordinate System (SPCS) uses LCC for most east-west-elongated states, and so do ICAO Operational Navigation Charts (ONC, 1:1,000,000), national weather charts, France's NTF and RGF93, and parts of Canada's topographic mapping. Two standard parallels (φ₁, φ₂) cut the cone into the Earth so that scale is below 1 between them and above 1 outside; placing the area of interest inside this band keeps distortion within ±1%.
n controls how open the projecting cone is. The cone, when unrolled into a plane, spans an arc of 2π·n radians. For a tangent cone (φ₁=φ₂), n = sin(φ₁): a low standard parallel gives a shallow cone (n → 0 approaches a cylinder, similar to Mercator), while a high one gives a sharp cone (n → 1 approaches the polar stereographic projection). LCC is the conformal projection that interpolates between Mercator and stereographic and is optimal for mid latitudes.
The scale factor k is the ratio of plotted distance to true ground distance. k=1 is exact, k=1.02 is 2% expansion, k=0.98 is 2% shrinkage. On a 1:50,000 aeronautical chart a 2% distortion places a point 200 m away from its true position over a 10 km leg — enough to fail instrument-approach tolerance. SPCS and ONC restrict the area covered by each chart so that |k-1| stays below ±0.05% near each standard parallel. This tool flags |distortion| ≤ 2% as warning and > 10% as failure.
Both are conformal (locally angle-preserving), but they target different bands. Mercator wraps a cylinder around the equator so it is accurate in the tropics but explodes the area of high-latitude regions (Greenland looks the size of Africa). The Lambert cone cuts the Earth at mid-latitudes, so distortion is minimised there. In aeronautical practice: Mercator for the tropics (South-East Asia routes), Lambert for the mid latitudes (North American and European charts), polar stereographic for polar routes.
Real-World Applications
ICAO ONC (1:1,000,000) international aeronautical charts: Almost every one of the roughly 270 sheets that cover the globe uses LCC. Each sheet covers a narrow latitude band (about 4° latitude × 6° longitude near the equator), with the two standard parallels placed about 1/6 of the way in from the top and bottom edges so that distortion stays inside ±0.05%. A pilot can lay a ruler on the chart and the distance read off is, for practical purposes, the true distance — which is exactly why LCC remains the basis for instrument approaches and en-route navigation worldwide.
US State Plane Coordinate System (SPCS): The state-plane grids for east-west elongated states (California, Texas, Colorado…) use LCC. Colorado Central, for example, sets φ₁=37°50′, φ₂=39°45′, λ₀=-105°30′; the scale factor stays in the 0.9999-1.0001 range everywhere inside the state. SPCS underpins cadastral surveying, road design, real-estate boundaries and the high-definition base maps used by autonomous-vehicle stacks. When combined with NAD83 the result is the de-facto US engineering coordinate system.
Operational weather charts: WMO upper-air analyses, 500 hPa charts and jet-stream maps for the mid latitudes are drawn on LCC grids. The shape of fronts and isobaric contours matters, so a conformal projection is preferred, and the area of interest (the westerly belt) lives exactly where LCC is most accurate. NCEP/GFS and ECMWF guidance products are output on internal LCC grids before being distributed to surface stations through national weather services.
French and Canadian national grids: France moved from the four-zone tangent-cone NTF to the modern RGF93 / Lambert-93 (secant, φ₁=44°, φ₂=49°, λ₀=3°, φ₀=46.5°), which covers all of metropolitan France in a single grid and is now the basis for IGN topographic maps, cadastre, car navigation and flood-hazard mapping. Parts of Canadian aeronautical and topographic mapping use LCC too. Whenever a GIS .prj file declares PROJCS["NTF (Paris) / Lambert …"], this is the projection in play.
Common Misconceptions and Pitfalls
First, the trap that "conformal means distance-correct". LCC preserves only local angles, not areas or distances. Because the scale factor k is latitude-dependent — for example k=1.005 at φ=30° and k=1.008 at φ=50° on the same chart — laying a ruler across the two points and reading the distance off the plane introduces an error of about 0.7%, or roughly 7 km over a 1,000 km leg. Mission planning and surveying must apply the latitude-dependent k correction. The "Distortion (%)" shown by this tool is the local scale error at the projection point, not the accumulated long-distance error.
Next, mixing spherical and ellipsoidal models. This tool uses a spherical Earth (radius R) to keep the maths transparent, but production LCC systems such as SPCS and RGF93 are defined on the GRS80 / WGS84 ellipsoid (semi-major axis a=6378137 m, flattening 1/298.257). The difference grows with latitude — at 45° N the two formulations disagree by about 0.3%. When using GIS software (QGIS, ArcGIS) or PROJ programmatically, always specify the projection through an EPSG code (e.g. EPSG:2154 for Lambert-93). For conceptual demos and teaching, the spherical version used here is perfectly adequate.
Finally, the singularity at φ=±90°. The LCC formulas include tan(π/4 + φ/2), which diverges at the pole and forces ρ → 0. This tool clamps the latitude sliders to ±89° to avoid the singularity, but if you implement LCC yourself you must guard against NaN and Infinity near the pole. Similarly, when the two standard parallels are almost identical (|φ₁ − φ₂| < 0.1°) the denominator of n becomes near-zero and unstable: branch and use the tangent-cone formula n = sin(φ₁) directly (this tool does exactly that).
How to Use
Set Standard Parallels (phi1, phi2): Enter two latitude values where the cone intersects Earth's surface—typically 30°N and 60°N for mid-latitude regions. These parallels have zero distortion.
Define Central Meridian (lam0): Specify the longitude (e.g., -95° for North America) where the projection is most accurate; x-coordinates extend east-west from this line.
Set Reference Latitude (phi0): Choose the latitude for false origin; combined with standard parallels, this governs the cone constant n and scale factor k.
Read outputs: Cone constant n determines convergence rate; Plane X/Y show projected coordinates in kilometers; Scale factor k and Distortion (%) indicate accuracy across your mapping region.
Worked Example
For a USGS topographic map covering the continental USA: Standard Parallels phi1=33°N, phi2=45°N; Central Meridian lam0=-96°W; Reference Latitude phi0=23°N. Output: Cone constant n≈0.629; for a point at 40°N, 90°W, Plane X=450 km, Plane Y=320 km; Scale factor k=0.998; Distortion=−0.2%; Cone-apex distance=3,850 km. Distances measured along standard parallels maintain scale; areas deviate by 0.2% across the zone, acceptable for 1:24,000 engineering maps.
Practical Notes
Choose standard parallels 1/6 to 1/5 of your region's north-south extent apart—wider spacing reduces overall distortion for large territories (e.g., Canada uses 49°N and 77°N).
Central Meridian should bisect your project area; off-center placement increases x-coordinate errors beyond ±500 km from the median line.
LCC preserves angles (conformal) but not areas—ideal for aeronautical charts, UTM transitions, and State Plane Coordinates; unsuitable for equal-area thematic mapping requiring areal accuracy.
Scale factor k varies with latitude; multiply ground distance by k to convert map distance, essential for surveying and GIS coordinate transformations.