Great circles and rhumb lines

How do you calculate the shortest spherical route?

A great circle is the intersection of a sphere with a plane through its centre.

Find the central angle

A great circle is formed by a plane passing through the centre of a sphere. Its shorter arc is the minimum surface distance between two non-antipodal points. On the training Earth, one degree of great-circle arc is 60 NM. This spherical convention is useful for navigation exercises; it is distinct from an ellipsoidal geodesic used by operational systems.

For latitudes φ1 and φ2 and the normalised longitude difference Δλ, cos σ = sin φ1 sin φ2 + cos φ1 cos φ2 cos Δλ. Distance is 60σ NM when σ is in degrees. A haversine or atan2 formulation improves numerical behaviour for very small distances and near antipodes. Keep degrees and radians consistent inside the calculation.

Initial track is local

From 50°N 020°W to 50°N 020°E, the spherical great-circle distance is approximately 1524.0 NM. The parallel route is 40 × 60 × cos 50° ≈ 1542.7 NM. The great circle bows toward the northern pole and its forward true direction changes, beginning near 074.42° and ending near 105.58°.

Initial track can be found with atan2(sin Δλ cos φ2, cos φ1 sin φ2 − sin φ1 cos φ2 cos Δλ), then normalised. The initial direction at departure is not a constant route heading. At antipodal endpoints many great circles have the same length, so an initial direction is not uniquely defined. At a pole, ordinary local true-azimuth conventions also need care.

Test the route choice

Compare both distances with the equal-latitude rhumb route before adding wind. A distance saving of a few nautical miles can be outweighed by a different headwind, a restricted area or an unavailable airway. The shortest geometric line is one input to the flight plan, not a complete optimum-time solution. In the animation, move the endpoints toward the equator: the great circle and parallel coincide there. Then move them north and increase longitude separation; both the bow and the difference in endpoint tracks grow. Explain those changes using the rotating local meridians rather than an imaginary turn at the centre of the map.

When is a constant true direction useful?

A rhumb line trades constant bearing against the shortest spherical path.

Constant bearing has a specific meaning

A rhumb line crosses each meridian at the same true angle. An east–west rhumb follows a parallel; a north–south rhumb follows a meridian. Only the equator and meridians are also great circles. A non-cardinal rhumb on a sphere spirals toward a pole without reaching it at a finite longitude.

On normal Mercator the rhumb is straight because the meridians are parallel and the projection preserves local angles. That plotting convenience does not establish minimum distance. Along a fixed latitude φ, the distance is 60 × longitude difference in degrees × cos φ NM. At 60°N, a 20° longitude rhumb therefore covers 600 NM.

Handle the general and limiting cases

A general spherical rhumb uses Δψ = ln[tan(45° + φ2/2) / tan(45° + φ1/2)] and q = Δφ/Δψ. With angles in radians, distance is R√(Δφ² + q²Δλ²); direction is atan2(Δλ, Δψ). For nearly equal latitudes, use the limiting value q = cos φ rather than dividing two numbers close to zero.

Normalise longitude difference to the intended crossing. From 179°E to 179°W, the short separation is 2°, although a route could deliberately take the long way if the problem specified it. Do not let a map edge turn a short date-line crossing into a 358° calculation.

A bearing is not a wind-corrected heading

Even if the rhumb track stays 090°T, the heading required to maintain it changes when wind changes. At TAS 120 kt with a 30 kt wind from north, the nose must point north of east; a southerly wind requires the opposite correction. A fixed track and fixed heading coincide only under suitable wind conditions. Use the route comparison laboratory to separate three ideas: the surface path, the north reference used to label it, and the velocity triangle needed to fly it. Each has its own calculation and source of error.

How does conversion angle connect a rhumb line and a great circle?

State whether the question asks for an approximation or an exact spherical result.

Connect the local directions

Conversion angle is the angle between a great-circle track and the rhumb-line track joining the same endpoints, at a specified endpoint. For a northern equal-latitude eastbound route, the rhumb track is 090° and the initial great-circle track is less than 090°. The great circle starts poleward of the rhumb and ends equatorward of it.

For a modest longitude span, an examination approximation often takes conversion angle as half the estimated Earth convergence: approximately ½ × longitude difference × sin mean latitude, with angles in degrees. It is an approximation, not an identity for arbitrary endpoints. Use the method explicitly required by the question; an exact trigonometric result can differ from a prescribed mental-dead-reckoning answer.

Compare an estimate with the spherical solution

For equal 50°N endpoints separated by 40° longitude, the approximate convergence is 40 sin 50° ≈ 30.64°, so estimated conversion angle is 15.32°. The exact initial spherical track is about 074.42°, giving an initial conversion angle about 15.58°. The difference is small here but grows as the approximation’s conditions weaken.

The exact change between initial and final forward great-circle tracks is not automatically the same as longitude difference multiplied by sine of one chosen latitude. Curvature and changing latitude matter. Distinguish Earth convergence, projection convergence and conversion angle in both the diagram and the arithmetic.

Invert the approximation

If a stated small-route exercise gives conversion angle 5° at mean latitude 30°, the approximate longitude span is 2×5/sin 30° = 20°. This inverse task checks what quantity the approximation represents. It does not determine a unique real route, because many endpoint pairs can have the same estimated conversion angle. In the laboratory, alter latitude and longitude span and compare the initial great-circle track with the fixed 090° equal-latitude rhumb track. Always name the endpoint at which the angle is being measured.

Can you explain and apply the mechanism?

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