Initial and Final Bearing on a Great Circle

A navigator asked “what bearing do I steer from JFK to London?” gets a more complicated answer than they might expect. “Initial bearing 51°” tells them where to start — but along the great-circle path, the bearing changes continuously. By the time the aircraft (or ship) reaches LHR, it's travelling on a bearing of 109°. The 58-degree deflection is a real feature of great-circle navigation, not a calculation error or a wind correction.

The great-circle-distance pillar covers the distance side. This article goes deeper on the bearing side: why two bearings exist, how to compute them, the true-vs-magnetic distinction, and the practical implications for navigation.

Why bearings change

A great circle on Earth (treated as a sphere) is the intersection of the Earth's surface with a plane through Earth's centre. Such a plane, except in the cases of the equator or a meridian, isn't aligned with Earth's rotation axis. As a navigator follows the great circle, the angle between the path and the local meridian (the local “north” direction) changes continuously.

Specifically:

• At the vertex of the great circle (the highest-latitude point on the path), the bearing is exactly 90° or 270° — the path is heading due east or due west, perpendicular to the local meridian.
• On either side of the vertex, the bearing tilts toward the poles. Approaching the destination from the vertex, the bearing rotates clockwise (if the path is in the northern hemisphere); leaving the start point toward the vertex, the bearing rotates counter-clockwise.

The net effect: for a path that crosses high latitudes, the initial and final bearings can differ by tens of degrees. For a path along the equator or along a meridian, the bearings are constant (the path is also a rhumb line).

The forward-azimuth formula

The initial bearing θ_initial from (φ₁, λ₁) to (φ₂, λ₂) on a sphere is computed as:

y = sin(Δλ) · cos(φ₂)
x = cos(φ₁) · sin(φ₂) − sin(φ₁) · cos(φ₂) · cos(Δλ)
θ_initial = atan2(y, x)

Where Δλ = λ₂ − λ₁ (longitude difference). The result is in radians; convert to degrees and normalise to [0, 360°) for compass-bearing reporting.

The formula uses atan2(y, x) rather than arctan(y/x) to handle the full circle of bearings correctly. The atan2 function returns values in [−π, π]; adding 2π and taking the modulo gives compass-convention bearings in [0, 360°).

The reverse-azimuth relationship

The final bearing at the destination is the direction the path is travelling when it arrives. It's the initial bearing from the destination back to the start, rotated 180° (because the path is heading toward the destination, not away from it):

θ_initial_reverse = atan2(sin(−Δλ) · cos(φ₁), cos(φ₂) · sin(φ₁) − sin(φ₂) · cos(φ₁) · cos(−Δλ))
θ_final = (θ_initial_reverse + π) mod 2π

Equivalently: compute the initial bearing from destination to start, then add 180°.

On a rhumb line, by contrast, the initial and final bearings are identical — the entire point of a rhumb line is that the bearing doesn't change. See /learn/what-is-a-rhumb-line for the constant-bearing case.

A worked example: JFK → LHR

Inputs:

JFK:  φ₁ = 40.6413°N,  λ₁ = −73.7781°W
LHR:  φ₂ = 51.4700°N,  λ₂ = −0.4543°W

Step 1: convert to radians and compute differences.

φ₁ = 0.7094 rad,  φ₂ = 0.8984 rad
λ₁ = −1.2876 rad, λ₂ = −0.00793 rad
Δλ = 1.2797 rad

Step 2: initial-bearing formula.

y = sin(1.2797) · cos(0.8984)
  = 0.9588 · 0.6228
  = 0.5972

x = cos(0.7094) · sin(0.8984) − sin(0.7094) · cos(0.8984) · cos(1.2797)
  = 0.7587 · 0.7825 − 0.6514 · 0.6228 · 0.2854
  = 0.5937 − 0.1158
  = 0.4779

θ_initial = atan2(0.5972, 0.4779) = 0.8954 rad = 51.30°

The initial bearing from JFK is approximately 51° — north-east. This matches the well-known fact that transatlantic flights from North America head north-east-ish before curving back south to European destinations.

Step 3: final-bearing formula (initial-from-LHR-to-JFK + 180°).

Δλ_reverse = −1.2797 rad
y_reverse = sin(−1.2797) · cos(0.7094) = −0.9588 · 0.7587 = −0.7274
x_reverse = cos(0.8984) · sin(0.7094) − sin(0.8984) · cos(0.7094) · cos(−1.2797)
          = 0.6228 · 0.6514 − 0.7825 · 0.7587 · 0.2854
          = 0.4057 − 0.1694
          = 0.2363
θ_initial_reverse = atan2(−0.7274, 0.2363) = −1.2580 rad = −72.06° → 287.94°
θ_final = (287.94° + 180°) mod 360° = 107.94°

The final bearing is approximately 108° — east-south-east. The arriving aircraft is heading roughly south-eastward when it reaches LHR, having curved over the Atlantic from its north-eastward initial heading.

The 57-degree difference between the initial and final bearings is the signature of the great-circle path's curvature in the north Atlantic.

True vs magnetic bearing

The formula above produces a true bearing — measured relative to true (geographic) north, the direction of the geographic North Pole. A compass needle, however, points to magnetic north — the direction of Earth's magnetic field at that location. The two differ by a quantity called the magnetic declination (or magnetic variation), which varies geographically and over time.

Sample declination values (approximate as of 2026, eastward positive):

• New York: −13° (compass needle points 13° west of true north)
• London: +1° (essentially aligned, recently changed sign)
• Honolulu: +10°
• Anchorage: +15°
• Wellington, NZ: +23°

For modern GPS-based navigation, the GPS receiver computes true bearings from coordinates and (usually) applies an internal World Magnetic Model (WMM) to display magnetic bearings on demand. For aircraft autopilots, the FMS converts as needed. For traditional compass navigation, the navigator manually corrects between true and magnetic using a published declination value for the area.

The WMM is the global standard for magnetic-field modelling; it's updated every five years (most recently 2025) and accounts for slow drift in the magnetic field. Some areas (especially near the magnetic poles) have rapidly changing declination — a single year can change it by 5–10°.

Practical implications

The change-of-bearing along a great circle has several practical consequences:

Manual navigation requires waypoint adjustment. A traditional navigator on a transatlantic flight or ship cannot simply set a single compass bearing and steer it — the bearing changes. Pre-GPS practice was to compute the great-circle path, divide it into segments, and recompute the bearing at each waypoint. Modern FMS / GPS computes the optimal bearing continuously.

Pilot reports use the initial bearing. When a pilot reports their planned heading at takeoff, they cite the initial bearing of the route. The final bearing isn't typically reported until the destination approach.

Antipodal paths have an undefined initial bearing. Two antipodal points (geometrically opposite on Earth) have no unique great-circle path — every meridian connects them. The formula above produces a numerical result, but it depends on floating-point details and isn't physically meaningful. For antipodal calculations, the bearing is conventionally taken as undefined.

The vertex bearing is informative. At the highest-latitude point of the great-circle path, the bearing is exactly 90° or 270°. Knowing where this vertex falls geographically gives a quick sense of how far north (or south) the path travels. For JFK → LHR, the vertex is at approximately 53°N (near the southern tip of Iceland), with the path bearing 90° due east at that point.

A second example: Sydney → Buenos Aires

A trans-Pacific great-circle path between two southern-hemisphere cities at similar latitudes:

Sydney (SYD):      φ₁ = −33.87°,  λ₁ = 151.21°
Buenos Aires (EZE): φ₂ = −34.61°,  λ₂ = −58.39°

Δλ = −58.39 − 151.21 = −209.60°
Normalised to (−180°, 180°]: Δλ = +150.40° (the eastward path is shorter)

Using the formulas:

y = sin(2.626 rad) · cos(−0.604 rad)
  = 0.4854 · 0.8228
  = 0.3994

x = cos(−0.591) · sin(−0.604) − sin(−0.591) · cos(−0.604) · cos(2.626)
  = 0.8302 · (−0.5685) − (−0.5575) · 0.8228 · (−0.8758)
  = −0.4719 − 0.4017
  = −0.8736

θ_initial = atan2(0.3994, −0.8736) = 2.713 rad = 155.4°

The initial bearing from Sydney is approximately 155° — south- south-east. The great-circle path passes south of New Zealand, then runs along the southern Pacific Ocean to South America. For the final bearing, an analogous calculation gives ~85° — nearly due east arriving at Buenos Aires.

The 70° bearing change reflects the path's southern detour: the great-circle path between two southern-mid-latitude cities arches south (toward the pole) because that's the shortest route. Just as transatlantic flights pass near Greenland (north detour), trans-Pacific southern flights pass near Antarctica's northern edge.

Bearing change rate

The bearing changes smoothly along the great-circle path — not in discrete steps. The instantaneous rate of change depends on the local latitude and the heading. Near the vertex (highest-latitude point), the bearing changes most rapidly per unit distance.

For aviation, the FMS computes the instantaneous required bearing along the route continuously and updates the autopilot every few seconds. For ship navigation under autopilot, the same approach applies. For manual navigation, waypoint-based segmenting is the practical approximation.

Common misconceptions

“Bearing is the same throughout a flight.” Only on rhumb lines (constant-bearing paths) and on a few special great circles (meridians, equator). On every other great-circle path, the bearing changes continuously.

“The initial bearing is the average bearing.” No — it's the bearing at the starting point. The average bearing along the great circle (if you wanted to compute one) would be the integrated mean bearing over the path; it can differ from the initial bearing by tens of degrees.

“True and magnetic bearings differ only at the magnetic pole.” They differ everywhere, by the local magnetic declination. In most populated regions the declination is a few degrees; near the magnetic poles it can exceed 100° or be effectively undefined.

“Bearing is computed differently for sphere vs ellipsoid.” The formulas differ in detail but not in spirit. Spherical formulas (above) treat Earth as a sphere; Vincenty's and Karney's ellipsoidal formulas produce slightly different bearings (typically within 1°) for the same coordinate pair on WGS 84. For most applications the spherical-formula bearing is sufficient.

“Magnetic north and the magnetic pole are the same.” The magnetic pole is the point on Earth's surface where the magnetic field points straight down. Magnetic north is the direction a horizontal compass needle points at a given location — which is toward the magnetic pole only at the equator; elsewhere, the magnetic field deviates more complicatedly. The WMM handles the full three-dimensional geometry.

“Bearings only matter on long flights.” They matter at all scales. Short-leg navigation (manually piloting a small boat in a harbour, hiking with map and compass) requires bearing computation even when the distance is small. The formulas scale down without modification.