ATPL General Navigation (061) cheat sheet

The Earth is modelled as an oblate spheroid: polar radius 6,357.0 km, equatorial radius 6,378.1 km, flattening 1/298.3; a sphere of radius 6,371 km is used for most navigation calculations.

Latitude is the angle between the equatorial plane and the local vertical (normal to the spheroid); geodetic latitude differs slightly from geocentric latitude, maximum difference about 11.5 arcminutes at 45 degrees.

1 minute of arc of latitude equals 1 nautical mile (1,852 m) by definition; a minute of arc of longitude equals 1 NM x cos(latitude).

Great circle: plane through the Earth's centre, shortest distance between two points, all meridians and the equator are great circles, and it cuts every meridian at a different angle.

Rhumb line: constant track crossing all meridians at the same angle; it is a loxodrome, spirals toward the pole, and on a Mercator chart plots as a straight line.

Position formats: latitude 0-90 N/S, longitude 0-180 E/W; convert with 1 degree = 60 minutes, 1 minute = 60 seconds; decimal degrees = degrees + minutes/60 + seconds/3600.

Change of latitude (ch lat) = difference in latitude in minutes; departure (dep) = ch long in minutes x cos(mean latitude); use these for dead reckoning and rhumb line legs.

Parallel of latitude: small circle (except the equator), constant latitude, all points on it have equal latitude but different longitude; meridians converge at the poles.

Antipodal point: latitude sign reversed, longitude = 180 - given longitude with E/W reversed; the shortest path between antipodes is 180 degrees of arc = 10,800 NM.

Magnetic variation (declination) is the angle between true north and magnetic north at a place; it is Easterly when magnetic north lies east of true north; True = Magnetic + Easterly variation (or Compass + deviation + variation).

Isogonals join points of equal variation; the agonic line joins points of zero variation; variation changes with position and slowly with time (annual change in minutes per year).

Deviation is the angle between magnetic north and compass north caused by aircraft/ship magnetism; it depends on heading and electrical/structural configuration, so a deviation card or table is required.

Compass conversion order: C D M V T (Compass to Deviation to Magnetic to Variation to True); reverse order T V M D C when converting true to compass.

Dip (inclination) is the angle of the Earth's magnetic field to the horizontal; it is 0 at the magnetic equator and 90 degrees at the magnetic poles; the directive force is horizontal component only.

The direct-reading (direct-indicating) magnetic compass is simple and independent of power but suffers from turning and acceleration errors; it is unreliable in turns and during acceleration/deceleration.

The indirect-reading compass system (flux valve/gyro-magnetic) uses a gyro stabilised heading with a flux valve slaving signal; it is more stable, but precession and apparent drift must be corrected.

Magnetic dip causes northerly/southerly turning error: in the Northern Hemisphere the compass under-reads on turns through north and over-reads through south; acceleration error shows a turn to the north during acceleration in the Northern Hemisphere.

Use the deviation card against the compass heading, not the magnetic heading, to obtain deviation; interpolate between tabulated headings.

Mercator projection: cylindrical, conformal (angles and shapes preserved), rhumb lines are straight lines, great circles are curved (concave toward the equator); scale increases with latitude by secant of latitude.

Transverse Mercator: cylinder axis in the equatorial plane, used for charts of north-south extent; Lambert conformal conic: cone projection, standard parallels, great circles nearly straight, used for many aeronautical charts.

Polar stereographic: conformal, used for polar areas; great circles are nearly straight lines through the pole.

Scale: representative fraction 1:500,000 means 1 unit on chart = 500,000 units on Earth; 1 NM = 1,852 m, so 1 cm on a 1:500,000 chart = 5 km = 2.70 NM.

Scale error: on a Mercator chart, scale at latitude L = scale at equator x sec(L); measure distance using the latitude scale at the same latitude as the leg.

Relief depiction: contour lines join equal elevation; hachures, layer tints, spot heights and shading show terrain; contour interval is the vertical distance between successive contours.

Chart properties to check: projection, scale, datum (WGS84), units of elevation, magnetic variation and its annual change, validity date, and whether it is a plotting or navigation chart.

On a Mercator chart, a straight line is a rhumb line; to plot a great circle, draw a straight line on a gnomonic chart or compute intermediate points and transfer them.

Large-scale charts (e.g. 1:250,000) show more detail over a smaller area; small-scale charts (e.g. 1:5,000,000) cover large areas with less detail.

Vector triangle: heading and true airspeed (TAS) is the air vector; wind direction and speed is the wind vector; resultant is track and ground speed (GS).

Wind direction is given as the direction the wind blows FROM; to add the wind vector, draw it in the direction the wind blows TO (add 180 degrees).

Track is the path over the ground; heading is the direction the aircraft points; drift = track minus heading (positive right/starboard drift); crab angle corrects for wind.

Formulas: GS = TAS x cos(drift) + wind component along track; crosswind component = wind speed x sin(angle between wind and track); headwind/tailwind component = wind speed x cos(angle).

The wind triangle can be solved by the sine rule: sin(drift)/wind speed = sin(wind angle)/TAS; use it to find drift and GS without a computer.

Dead reckoning position uses heading, TAS, elapsed time and wind: distance = GS x time; ch lat = distance x cos(track); departure = distance x sin(track); ch long = departure / cos(mean lat).

1 in 60 rule: a 1 degree error over 60 NM gives 1 NM off track; use it for off-track correction and for drift estimation.

Time = distance / GS; fuel = fuel flow x time; always convert minutes to decimal hours (e.g. 12 min = 0.2 h) before multiplying.

Wind velocity (W/V) is often given in degrees true and knots; convert to magnetic when working with magnetic headings using variation.

Visual navigation: identify ground features (towns, roads, rivers, railways, coastlines) and compare with the chart; use checkpoints, timing and heading corrections.

Map reading: orient the chart with north up or with track up; use the scale and latitude scale for distance; account for chart datum and magnetic variation.

Position line (PL): a line of constant bearing or distance from a known feature; a fix requires at least two position lines (three for a reliable fix) intersecting at a small angle (ideally 60-90 degrees).

VOR gives a magnetic bearing (radial) from the station; a VOR position line is a straight line from the station; two VOR radials give a fix; DME gives slant range, which is greater than ground distance at high altitude.

NDB/ADF gives a relative bearing; true bearing = magnetic heading + relative bearing (then apply variation); ADF is affected by coastal refraction and night effect.

ILS gives guidance only for approach; the localiser gives lateral guidance and the glide path gives vertical guidance; not used for en-route fixes.

Radar and GNSS position lines: GNSS gives a direct position fix; radar gives bearing and distance; use the best geometry and cross-check sources.

Track error angle = difference between desired track and actual track; use the 1 in 60 rule: correction = (distance off track / distance flown) x 60 degrees.

Closing angle to destination = (distance off track / distance to go) x 60 degrees; total heading change = track error + closing angle.

Leg time = leg distance / GS; leg fuel = leg time x fuel flow; sum legs for total trip fuel; add reserves (contingency, alternate, final reserve) as required.

Fuel flow is often given in kg/h or litres/h; convert units consistently (fuel density about 0.72-0.8 kg/l for jet fuel) before comparing.

Point of no return (PNR): distance to PNR = (total fuel available x GS out) / (GS out + GS back), using fuel available excluding reserves.

Point of equal time (PET/CP): distance from departure = (total distance x GS out) / (GS out + GS back); the point where time to either destination is equal.

Replanning in flight: recalculate GS with the actual wind, update ETA and fuel remaining; if headwind increases, PNR moves closer to departure.

Use the triangle of velocities to update GS: GS = TAS x cos(drift) + wind component; a headwind reduces GS and increases leg time.

Critical fuel: if fuel remaining at destination falls below reserves, divert or reduce power; compute fuel required to alternate = distance / GS x fuel flow.

Time conversions: UTC is the reference; LMT = UTC + longitude/15 (east positive); standard time = UTC + zone offset; zone boundaries may not follow meridians exactly.

Great circle distance: use the spherical law of cosines: cos(D) = sin(lat1)sin(lat2) + cos(lat1)cos(lat2)cos(ch long); D in degrees x 60 = distance in NM.

Initial great circle bearing: tan(initial bearing) = sin(ch long) / (cos(lat1)tan(lat2) - sin(lat1)cos(ch long)); convert to true bearing.

Great circle track crosses each meridian at a different angle; the track reaches its highest latitude at the vertex, where the track is perpendicular to the meridian.

Rhumb line distance: use departure = ch long x cos(mean lat), then distance = sqrt(ch lat^2 + departure^2); for long legs use Mercator sailing with meridional parts.

Rhumb line bearing: tan(course) = departure / ch lat; the rhumb line is a straight line on a Mercator chart and has constant track.

Great circle vs rhumb line: the great circle is shorter except along the equator or a meridian; the rhumb line is easier to steer but longer.

Plotting: on a Mercator chart, plot a great circle by computing intermediate points or by using a gnomonic chart where great circles are straight lines.

Composite great circle: when the great circle would go above a limiting latitude, follow a great circle to the limiting parallel, then a parallel, then a great circle; used to avoid polar areas.

Vertex latitude: cos(vertex lat) = cos(lat1) x sin(initial bearing); use to check the highest latitude reached.

UTC is the reference time; it is based on atomic time and kept within 0.9 s of UT1 by leap seconds; aviation uses UTC for all flight planning and position reporting.

LMT = UTC + longitude/15 hours (east positive, west negative); 15 degrees of longitude = 1 hour; 1 degree = 4 minutes.

Standard time = UTC + zone offset; zone offsets are usually whole hours but may be half or quarter hours (e.g. India UTC+5:30, Nepal UTC+5:45).

Zone boundaries follow political borders, not meridians; always check the actual zone offset for the location rather than assuming longitude/15.

Sunrise/sunset: LMT of sunrise = 12:00 - (day length/2) adjusted for latitude and season; use tables or approximate with the equation of time.

Daylight saving time adds 1 hour to standard time; convert to UTC by subtracting the offset, and check whether DST is in force.

Crossing the date line (180 degrees) changes the date by one day; going east across the line subtracts a day, going west adds a day.

Time problems: elapsed time = UTC arrival - UTC departure; if the result is negative, add 24 hours and adjust the date.

RNAV allows navigation on any desired path without being constrained to pass over ground-based stations; it uses VOR/DME, DME/DME, GNSS or inertial inputs.

GNSS position is determined by trilateration from at least 4 satellites (3 for position, 4 to resolve clock bias); the receiver measures signal travel time and computes pseudoranges.

GPS constellation: about 24-31 satellites in 6 orbital planes at about 20,200 km altitude, 55 degree inclination, orbital period about 12 hours.

Accuracy: GPS standard positioning service about 5-15 m horizontal; augmentation (SBAS, GBAS) improves accuracy and integrity; RAIM provides integrity monitoring.

RNAV route names and waypoints: 5-letter waypoints, RNAV airways, and PBN specifications (RNAV 5, RNAV 1, RNP 1); RNP requires on-board performance monitoring and alerting.

Required navigation performance (RNP) includes a containment region; RNP 1 means total system error must be within 1 NM for 95% of the flight time.

GNSS errors: ionospheric and tropospheric delay, multipath, satellite clock and ephemeris errors; RAIM detects faults but requires sufficient satellites in view.

Waypoint coordinates and datum: use WGS84 for GNSS; chart datum differences can cause position errors if not converted.

RNAV leg types include track-to-fix, direct-to, and radius-to-fix; the FMS computes the path and provides guidance and autopilot coupling.