Great Circles and Rhumb Lines: Properties, Calculations and Plotting
Properties
A great circle is the intersection of the Earth’s surface with a plane through the centre; it is the shortest surface distance between two points. The equator and all meridian pairs are great circles. A rhumb line cuts every meridian at the same angle and has constant track; it is longer than the great circle except on the equator or a meridian. On a Mercator chart rhumb lines are straight and great circles curve toward the pole; on a gnomonic chart great circles are straight; on Lambert conformal conic great circles are nearly straight.
Calculations
For a great-circle distance use the spherical cosine formula: cos D = sin φ1 sin φ2 + cos φ1 cos φ2 cos Δλ, where D is angular distance; multiply by 60 for NM. Initial track requires the appropriate spherical formula or a chart/gnomonic plot. The vertex is the highest latitude reached. Rhumb-line distance for an east-west leg is departure = Δλ (minutes) × cos φ; for a general rhumb line use Mercator sailing. Great-circle legs are normally broken into rhumb-line segments for plotting.
Worked example
From 30°N 000° to 30°N 060°E. Rhumb distance = 60° × 60 × cos 30° = 3 117.7 NM. Great-circle: cos D = sin²30 + cos²30 cos60 = 0.25 + 0.75 × 0.5 = 0.625; D = 51.32°; distance = 51.32 × 60 ≈ 3 079 NM. The great circle is about 39 NM shorter. Its track starts northeast and reaches a vertex north of 30°N before turning southeast.
Traps
Do not use departure = Δλ cos φ for a long north-south or diagonal great-circle leg. Do not call a straight Mercator line the shortest route. Watch the anti-meridian when Δλ is near 180°. Remember that initial great-circle track differs from the final track. Plot great circles on a gnomonic chart or with waypoints, not as a single straight line on Mercator.