Why Conical Charts?

The Mercator chart has two important limitations for modern aviation:

• Great Circles are not straight lines — they appear as curves concave to the Equator. Modern aircraft with FMS/INS fly Great Circle tracks, and a chart where the planned track appears curved is inconvenient.
• Scale is not constant — scale changes significantly with latitude on a Mercator chart, making it impossible to measure distances with a ruler. Instead, dividers must be used, adjusted to different sizes at different latitudes.

After 1960, with the advent of automatic computing (INS, IRS, FMS, GPS), it became practical to fly Great Circle tracks. A chart on which a Great Circle appears as a straight line is therefore far more useful for route planning.

The solution: the Lambert's Conical Conformal Chart, developed in 1777 by Swiss mathematician Johann Heinrich Lambert.

The Simple Conical Projection

The basis of the Lambert chart is the simple conic. A light source at the centre of the Reduced Earth casts shadows of the graticule onto the inside surface of a cone placed over the globe. The cone touches the globe at the parallel of tangency. The cone is then slit and opened flat.

Figure 1 — Principle of simple conical projection — light source at the centre projects the graticule onto the cone; scale is correct at the parallel of tangency

Figure 2 — Simple conical graticule — when the cone is slit and opened flat, 360° of longitude appears as a sector of a circle; meridians radiate from the pole; parallels are arcs of concentric circles

Figure 3 — Apex angle is twice the parallel of tangency: at 45°N tangency, the apex angle = 90°; at 60°N, apex = 120°; at 90°N (flat sheet), apex = 180°

The Arc of Sector Formula

When the cone is opened flat, 360° of longitude is represented by an arc of less than 360°. The size of this arc depends on the parallel of tangency:

Examples:

• Parallel of tangency 45°N: arc = 360° × sin(45°) = 360° × 0.707 = 255°. Missing sector = 105°.
• Parallel of tangency 60°N: arc = 360° × sin(60°) = 360° × 0.866 = 312°. Missing sector = 48°.
• Parallel of tangency 90°N: arc = 360° × sin(90°) = 360° × 1.0 = 360°. No missing sector (flat sheet — the Polar Stereographic case).

The relationship between 360° of longitude and the arc of the sector is called the constant of the cone, denoted n:

Figure 4 — Change of longitude vs chart convergence: 100° ch.long is represented by a smaller angle on the chart — the ratio depends on n (sin of parallel of tangency)

Figure 5 — Chart and Earth meridian angles compared — on the chart the angle between meridians is less than the Earth angle between the same meridians

Scale on the Simple Conic Projection

On the simple conic, scale is correct only at the parallel of tangency. It expands away from this parallel in both directions. The expansion is quite rapid — much faster than on a Lambert chart.

Figure 6 — Scale on the simple conic: correct only at the parallel of tangency (shown in green); expands away from it toward higher and lower latitudes

Figure 7 — Illustrating scale expansion on the simple conic — the expanding spacing of the parallels shows how scale grows away from the tangent parallel

Lambert's Modification — Bringing the Cone Inside

Lambert's key insight: by pushing the cone inside the Reduced Earth (rather than leaving it touching the outside), the single line of correct scale at the parallel of tangency is replaced by two parallels of correct scale — the standard parallels. Between the standard parallels, scale is slightly compressed; outside them, scale is slightly expanded.

The result is a chart with near-constant scale across the whole projection, making distance measurement with a ruler practical.

Figure 8 — Lambert step 1: simple conic showing scale correct at tangent parallel, expanding rapidly outside

Figure 9 — Lambert step 2: cone pushed inside the RE — the single tangent parallel becomes two secant parallels (standard parallels)

Figure 10 — Lambert modification — the cone now intersects the Reduced Earth at two latitudes (standard parallels) rather than touching at one

Figure 11 — Lambert scale zones: scale contracts between the standard parallels (blue, scale < 1), is correct AT the standard parallels (green), and expands outside (yellow/red)

Figure 12 — Lambert scale zones continued — the expanded and contracted zones are exaggerated here to show the principle; in practice the variation is tiny across the chart

Non-perspective Chart

Having brought the cone inside the Reduced Earth, Lambert made mathematical adjustments to ensure the chart is orthomorphic. The result is that the Lambert chart is a non-perspective chart — it cannot be produced by a simple optical projection. It is constructed mathematically.

This is the same situation as the Mercator: the mathematical modification required to achieve conformality removes the direct optical projection relationship.

Figure 13 — Part of a Lambert conical orthomorphic chart — the familiar look of the ICAO 1:500 000 or 1:250 000 aeronautical chart; note slightly curved parallels and converging meridians

Figure 14 — Orthomorphism on the Lambert chart: meridians converge (left) and parallels curve (right), but when combined they intersect at exactly 90° — satisfying both orthomorphism conditions

Lambert Chart Properties

Scale

Scale is least at the Parallel of Origin (between the two standard parallels). It increases outward, reaching the correct value at the two standard parallels. Scale is greatest at the top and bottom parallels of the projection.

Parallel of Origin

The Parallel of Origin is the mathematical basis of the projection. It is assumed to be halfway between the two standard parallels. Its sine is the constant of the cone (n):

Figure 15 — The Parallel of Origin — the mathematical basis of the Lambert projection; it lies halfway between the two standard parallels and defines the convergence factor n

Figure 16 — Chart convergence on the Lambert chart is constant across the whole chart — all meridians incline at the same angle to each other

Graticule

• Meridians: Straight lines radiating from the pole (pole is usually off the map)
• Parallels: Arcs of concentric circles centred at the pole
• Meridians and parallels intersect at 90°

Chart Convergence

Since the meridians on the Lambert chart are straight lines, the angle between any two given meridians does not change with latitude — unlike on the real Earth where meridians converge increasingly toward the poles.

This is different from Earth convergency which = ch.long × sin(latitude) and changes with latitude. Chart convergence on the Lambert is correct only at the Parallel of Origin. Above the Parallel of Origin, chart convergence < Earth convergency; below it, chart convergence > Earth convergency.

Rhumb Lines and Great Circles

Rhumb Lines

Except for meridians (which are straight lines), rhumb lines on a Lambert chart appear as curves concave to the pole (i.e. they curve in the same direction as the parallels of latitude). The curvature is considerable.

Great Circles

Figure 17 — Great circles on the real Earth — three Great Circle tracks at different latitudes, shown as straight lines across the globe

Figure 18 — Great circles on a Lambert chart — when Earth meridians are straightened to parallel-line chart meridians, the formerly straight GCs become slightly curved; the curvature is concave to the Parallel of Origin

Figure 19 — Earth convergence vs chart convergence — the Lambert chart convergence is fixed at the Parallel of Origin value; at the Parallel of Origin, chart = Earth, and a Great Circle is a straight line