Mercator scale, chart measurement and the great-circle decision
During a North Atlantic flight-planning exercise you use a Mercator chart whose stated scale is 1:5,000,000 at the equator. You measure 40.0 cm between two waypoints that both lie on the 60°N parallel. The aircraft TAS is 460 kt and the operator's cost figure is 1,150 kg of fuel per 100 NM. The great circle joining those two waypoints lies poleward of the parallel, whereas the straight line you measured on the chart follows the parallel. A colleague proposes using the chart measurement directly for the flight plan without correcting the scale. You also have a second leg at 40°N where the chart distance is 14.0 cm; the same colleague treats it as if the chart scale were still 1:5,000,000 at that latitude. You must decide which distance is credible and how much fuel the great-circle route saves if it is 3% shorter than the chart-line distance.
1. Calculate the true distance represented by the 40.0 cm measurement at 60°N.
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On a Mercator chart the scale varies with latitude. The scale factor is sec φ, so the representative fraction at latitude φ is the equator scale divided by cos φ. At 60°N, cos 60° = 0.5, so the scale is 1:5,000,000 × 0.5, i.e. 1:2,500,000. A chart length of 40.0 cm therefore represents 40.0 × 2,500,000 = 100,000,000 cm on the Earth, which is 1,000 km. Dividing by 1.852 gives 540 NM. Using the equator scale uncorrected would give 1,080 NM, exactly twice the true value. The chart measurement is a distance along the 60°N parallel, not a great-circle distance, so 540 NM is the rhumb-line distance between the two waypoints, not the shortest surface distance.
2. Explain why the straight line measured on the chart is not the shortest path and why the great circle appears poleward of the parallel.
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A straight line on a Mercator chart is a rhumb line: it crosses every meridian at the same angle and therefore represents a constant true track. It is not a great circle except along the equator or a meridian. The great circle is the shortest path on the sphere, and on a Mercator chart it plots as a curve. For two points in the northern hemisphere, that curve lies poleward of the rhumb line because the great circle takes advantage of the convergence of the meridians at higher latitude. The 60°N parallel is itself a rhumb line, so measuring along it gives the longer, constant-heading distance. The chart cannot show the great-circle length directly as a straight-line measurement at constant scale; the flight-plan distance must come from a great-circle calculation or from waypoints along the great-circle track.
3. Using the 540 NM chart-line distance and a 3% great-circle saving, calculate the fuel saved and comment on whether the colleague's uncorrected scale changes the decision.
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A 3% saving on 540 NM is 0.03 × 540 = 16.2 NM. At the stated cost of 1,150 kg per 100 NM, the saving is 16.2 × 11.5 = 186 kg. If the colleague had used the uncorrected equator scale, the distance would have been 1,080 NM and the apparent 3% saving would be 32.4 NM and 373 kg. The decision to flight-plan the great circle does not change, but the fuel figure is doubled and therefore misleading. The correct fuel saving is about 186 kg, based on the true rhumb distance of 540 NM. The great circle remains the shorter route; the chart-line distance is only a valid rhumb-line measurement after the secant-latitude scale correction.