ATPL general navigation course: 14 illustrated chapters

The surface of the oceans at rest (the geoid) is not a sphere: the Earth's rotation flattens it at the poles and makes it bulge at the equator. It is approximated by an ellipsoid of revolution; the reference one is WGS 84, used by GPS and for publishing aeronautical coordinates.

For everyday navigation calculations, the Earth is considered a perfect sphere with a circumference of 40,000 km, i.e. 21,600 NM (360° × 60').

On an ellipsoid, the vertical at a place (perpendicular to the local horizon) does not, in general, pass through the centre of the Earth.

The two coincide at the equator and at the poles; the difference, very small (about 11'), is greatest around 45° N and S. A consequence of the flattening: the real length of one minute of arc of meridian varies slightly (1,843 m at the equator, 1,862 m at the pole) and is 1,852 m around 45°.

A point is fully defined by two coordinates (WGS 84 reference).

The poles are the points where the axis of rotation pierces the surface. Meridians are parallel to one another only at the equator; they converge towards the poles.

Difference of latitude ℓ: same hemisphere, subtract; opposite hemispheres, add. It is named N or S according to the direction of movement.

Difference of longitude g: same side of Greenwich, subtract; opposite sides, add. If the result exceeds 180°, take 360° − g: the shortest path then passes through the antimeridian. It is named E or W according to the direction.

Examples:

56° N 145° E → 57° N 165° W: g = 35° + 15° = 50° towards the East;

01°15' N 090° E → 015°15' E: g = 74°45' W;

01°15' N → South Pole: ℓ = 91°15' S.

The antipode of a point has the latitude of opposite name and the longitude 180° − G of opposite name: 48°50' N 002°16.5' E → 48°50' S 177°43.5' W.

The nautical mile (NM) is the length of an arc of a great circle subtending an angle of 1 minute at the centre of the Earth: 1' = 1 NM, 1° = 60 NM. Its international (ICAO) value is 1,852 m, the mean length of one minute of meridian (exact at about 45° latitude).

Since meridians are great circles, a North-South distance can be read directly: distance (NM) = difference of latitude in minutes.

The kilometre was defined as 1/10,000 of the distance from pole to equator (hence 40,000 km around the Earth).

A parallel of latitude L is a circle of radius R cos L. One minute of longitude there therefore measures cos L nautical miles. The East-West distance between two meridians, along a parallel, is called the departure (or East-West distance):

e = g × cos L (g in minutes, e in NM).

It decreases as latitude increases: at 60°, 1° of longitude is worth only 30 NM and the full circle 10,800 NM. An aircraft flying around the world at 60° N at 480 kt would take the same time as at 960 kt on the equator.

Conversely, for a flight due East or due West: g = e / cos L. Example: 165 km = 89.1 NM towards the West at 60° N gives g = 178' = 2°58'.

For a short leg following a true track Tt over a distance D, break it down as follows:

Example in the figure: from the equator, 200 NM on 240° leads to 01°40' S 097°07' E. Another case: 100 NM on 225° from 00° 130° E: ℓ = e = 70.7 → 01°11' S 128°49' E.

Classic pitfalls: an aircraft leaving the equator due South for 10 h at 540 kt covers 5,400 NM = 90°: it reaches the South Pole. The square "600 NM South, East, North, West" from 04° N 170° W does not close: the East leg is flown at 06° S (600 / cos 6° = 603.3' of longitude), the West leg at 04° N (600 / cos 4° = 601.5'). 1.8' remains towards the East: arrival at 04°00' N 169°58.2' W.

International units: horizontal distances in m, km and NM; vertical distances in m and ft.

Typical calculations:

9.5 km = 9,500 / 0.3048 = 31,168 ft;

730 ft/min = 730 × 0.3048 / 60 = 3.7 m/s;

70 m/s = 70 × 3,600 / 1,852 = 136 kt;

36.25 in = 92.08 cm;

100 SM in 20 min = 300 SM/h = 260.7 kt: 215 NM take 49.5 min;

840 NM in 132.4 min = 380.7 kt = 705 km/h;

526 km/h = 284 kt: 284 NM in 1 h.

With the exam calculator, beware: its default "mile" is the statute mile (SM), not the nautical mile.

Two points located on opposite meridians (longitudes 180° apart) are joined by a great circle passing through the pole: the distance is calculated as a difference of latitude.

Flights of the same length do not cover the same longitude depending on latitude. Two aircraft leave Greenwich on 090° for 1,200 NM, one on the equator (20°), the other on the 45° N parallel (1,200/0.707 = 1,697' = 28.3°): on arrival they are separated by about 8° of longitude.

The Sun travels 15° of longitude per hour: to keep it due South while following 60° N, you need 15 × 60 × 0.5 = 450 kt towards the West.