Distance, departure and direction

How does an angle become a distance?

Choose the circle along which the distance is measured.

Latitude and longitude do not use one universal ruler

For examination-scale spherical calculations, a minute of arc along a great circle is 1 NM. A meridian is part of a great circle, so 2°30′ change in latitude on one meridian is 150 NM. The international nautical mile is exactly 1,852 m; a knot is 1 NM per hour.

A parallel away from the equator is smaller. For travel along a parallel, departure = longitude difference in minutes × cos latitude. At 60°N, 5° longitude is 300′ × 0.5 = 150 NM. This is the distance along the parallel, not generally the shortest distance between the endpoints.

Invert the calculation deliberately

To find longitude change from departure, divide by 60 cos latitude when using degrees. A 120 NM eastward displacement along 60°N gives 120/(60×0.5)=4° longitude east. Near a pole the denominator approaches zero: longitude becomes a poor local distance coordinate.

A flat north/east approximation can combine small latitude and departure changes using Pythagoras. Use mean latitude only for appropriately short legs. Across long legs, high latitudes or the date line, use spherical or ellipsoidal methods and normalise longitude differences.

Choose the approximation before calculating

Consider a local displacement of 60 NM north and 60 NM east near 45°N. A north/east plane gives distance √(60² + 60²) = 84.85 NM and track 045°. The latitude change is about 1°; using a mean latitude near 45.5° gives a longitude change of approximately 1.43°. This checks the scale of a chart plot.

The same method becomes less reliable as the leg grows because the local north/east frame rotates and latitude changes along the path. A 60 NM displacement near a pole can span a large longitude angle even though its distance is modest. State whether the problem asks for a distance along a meridian, along a parallel, along a rhumb line or along the shortest great-circle route before selecting a formula.

An air mile measures motion through the air

Air distance uses TAS and elapsed time, rather than ground speed. At TAS 180 kt for twenty minutes, air distance is 60 nautical air miles under the constant-speed assumption. If a direct headwind is 30 kt, GS is 150 kt and ground distance over the same time is only 50 NM. The two distances describe different reference frames.

A performance or fuel exercise may define an air-mile convention or equivalent still-air distance. Read that definition before substituting a ground-route distance. Wind does not directly change the aircraft’s air distance in this simple case, but it changes how much ground route that air motion completes.

What is the reference for a direction?

A direction is incomplete until its north reference and sense are known.

Track, heading and bearing describe different lines

Heading is where the aircraft’s longitudinal axis points. Track is the direction of motion over the ground. Bearing is the direction from one point to another. A crosswind can make heading and track differ while the aircraft flies without sideslip.

True directions use the local true meridian. Magnetic directions use magnetic north. Compass directions include the aircraft compass’s deviation. Grid directions use grid north on the selected chart. Convert references before subtracting directions or comparing a chart track with an instrument indication.

Wrap angles before choosing a turn

Directions are cyclic. 359° to 001° is a 2° clockwise change, not 358° anticlockwise. The reciprocal of 025° is 205°; of 220° is 040°. Add or subtract 180° and reduce to 000–359°.

A great-circle track’s true direction usually changes along the route because local meridians converge. The reciprocal of an initial great-circle track is the reverse direction at that same point; it is not automatically the initial track when departing the far endpoint.

Test a reciprocal at both ends

Imagine two nearby fixes with a straight local track of 035°T from A to B. At B, the direction back to A is approximately 215°T in the local-plane model. If the chart or receiver instead supplies a magnetic bearing, convert it using the local variation before comparing it with the true track. Agreement between two different north references can be accidental.

For a long great-circle leg, calculate the reverse initial track at B separately. Local north has rotated relative to the great-circle plane during the journey. Adding 180° to the departure direction gives a reciprocal at A; it does not transport that direction to B. This is why a route can be shortest and still require different forward true tracks at its endpoints.

Can you explain and apply the mechanism?

Four original questions. Work from the stated facts, then compare your reasoning with the explanation.