Introduction

The earliest method of transferring the graticule of meridians and parallels from a globe to a flat chart was the simple cylindrical projection. The cylinder touches the globe at the Equator. Projecting the graticule onto this cylinder and rolling it flat gives the result shown in Figure 1 — notice that the shapes are stretched progressively more toward the poles.

Figure 1 — Simple cylindrical projection: when the cylinder is unrolled, the parallels are equally spaced but the meridians are parallel vertical lines — shapes stretch increasingly toward the poles (Mercator's starting point)

The 16th-century Flemish cartographer Gerardus Mercator identified the problem: the N-S scale on this simple chart changes (expands toward the poles) at a different rate to the E-W scale. This violates Condition 2 for orthomorphism. He mathematically adjusted the spacing of the parallels so that, at any point, the N-S scale equals the E-W scale. The result is the famous Mercator Chart.

Scale Expansion on the Mercator Chart

To make the chart orthomorphic, Mercator increased the spacing of parallels toward the poles. This means that although the chart is now conformal, scale expands with increasing latitude.

Figure 2 — Mercator scale expansion: the N-S spacing of parallels increases with latitude to maintain equal scale in all directions at each point — this is the price of orthomorphism

The scale expansion factor is the secant of the latitude:

The 8° ≈ 500 NM Approximation

In examination questions, 8° of latitude ≈ 500 nautical miles. This is a useful shortcut — when you see the number 500 NM in a Mercator scale question, it almost certainly refers to 8° of latitude change.

The reason: 8° × 60 NM/° = 480 NM ≈ 500 NM at the Equator (where 1 minute of arc = 1 NM). The approximation works well for small scale changes at moderate latitudes.

Figure 3 — The Mercator Projection — scale expands away from the Equator, most dramatically at high latitudes; the chart is conformal at every point but cannot show the poles

The Secant Formula and Orthomorphism

The mathematical definition of the secant: sec(θ) = 1/cos(θ). This means scale expansion on a Mercator chart is:

• At 0° (Equator): sec(0°) = 1 — correct scale
• At 30°: sec(30°) = 1/cos(30°) = 1/0.866 = 1.155 — 15.5% expansion
• At 60°: sec(60°) = 1/cos(60°) = 1/0.5 = 2.0 — scale doubled
• At 90° (Poles): sec(90°) = 1/cos(90°) = 1/0 = ∞ — the poles cannot be shown

Figure 4 — Scale expansion examples on a Mercator chart — shapes in the same latitudinal band appear correct relative to each other, but are stretched compared with true scale

Figure 5 — (a) Convergent projection (approximately correct shape) vs (b) Mercator projection (stretched): on the Mercator all meridians are parallel vertical lines; on a convergent chart they converge toward the poles

Rhumb Lines on a Mercator Chart

The most powerful property of the Mercator chart for traditional navigation:

Because meridians are parallel vertical lines on the Mercator, any constant compass bearing crosses every meridian at the same angle — which is exactly what a straight line does on this chart. A pilot can therefore draw a straight line on a Mercator chart, measure its angle to a meridian, and that angle is the constant heading to fly.

This property made the Mercator chart the dominant navigation chart from the 16th century until INS/GPS computing became commonplace after ~1960.

Great Circles on a Mercator Chart

Figure 6 — Rhumb line vs Great Circle — the Great Circle (shortest distance) curves away from the Rhumb Line, appearing concave toward the Equator when viewed from above the Equator

Because meridians converge on the Earth but are parallel on a Mercator chart, great circles — which are straight on the globe — must be curved on the Mercator:

• The Equator: a great circle that appears as a straight line (the line of tangency)
• Meridians: great circles that appear as straight vertical lines
• All other great circles: appear as curves concave toward the Equator

Figure 7 — Mercator chart showing rhumb lines and great circles between major cities: the RL between London and LA is approximately 257°T (straight line); the GC track curves considerably northward

Figure 8 — Comparison of RL track (blue, straight) and GC track (red, curved) on a Mercator projection — the GC is concave to the Equator and shorter in distance but harder to fly without FMS computing

The Conversion Angle — Straight Line vs Great Circle

The angle between a straight line drawn on a Mercator chart and the corresponding Great Circle is equal to the Conversion Angle (CA):

Direction of correction:

• Northern hemisphere, eastbound: GC initial track < RL track (GC is more northerly at departure)
• Northern hemisphere, westbound: GC initial track > RL track
• Southern hemisphere: reverse the above

Alternatively: the GC initial track is always closer to the nearer pole than the RL track. In the Northern Hemisphere, "closer to the North Pole" means a smaller track angle for eastbound legs and a larger track angle for westbound legs.

Summary of Mercator Properties

Practice Questions