Lambert conformal conic

Navigation · Chapter 12 · DGCA CPL and ATPL

Why many charts are made on a cone, where their scale is right, how much the meridians converge, and how routes look on them.

Remember

Flattened cone: sector = 360° × n; n = sin(parallel of origin).
Origin = mean of the two standard parallels; one-sixth rule; 16° apart ≈ 1 % error.
Scale correct on the standard parallels, least at the origin, expanded outside.
Chart convergence = n × ch long, the same everywhere.
Straight line ≈ great circle; rhumb line concave to the pole; measure at the mid-meridian.

A cone on the globe

What pilots want • Great circles straight (shortest routes; INS and GPS fly them) • Scale nearly constant • Mercator gives neither; a cone does better
Cone angle = 2 × latitude • The cone touches the globe along 45°N • Its sides meet at the apex at 2 × 45 = 90°
Cut it and open it • Laid flat, the cone is a sector, not a full circle • Its share of 360° is the constant of the cone, n • Here: 255° out of 360° → n = 255 ÷ 360 = 0.71
Push the cone down • Now it cuts the globe along two standard parallels: 36°N and 54°N (gold) • Half-way, 45°N, is the parallel of origin: n = sin 45° still
Open the cut cone • Scale is correct on the gold lines • Least between them, expanded outside
Why "conformal" • Meridians are straight lines from the apex; parallels are arcs round it • They cross at 90°, and at any point the scale is the same every way: bearings are true

Any latitude

Flatten the cone • Cut it open: a sector • Touching at 45°N: 360 × sin 45° = 255°
Move the latitude • Near the equator: a thin sector • At the pole: a whole circle (a flat sheet)
The constant of the cone • n = sin(latitude) • It is the fraction of 360° the sector keeps