Comparison of Rhumb Lines and Great Circles

A line in a plane has constant slope and represents the shortest path between two points. A line on a sphere can have constant slope or can represent the shortest path between two points, but not necessarily both. If a line on a sphere has constant slope (constant azimuth), then the line follows a rhumb line path. If a line on a sphere represents the shortest path between two points, then the line follows a great circle path.

This figure illustrates two distant locations connected by a great circle path and a rhumb line path.

Rhumb Lines

A rhumb line, also known as a loxodrome, is a curve with a constant azimuth. An azimuth is the angle a line makes with a meridian, measured clockwise from north.

All parallels are rhumb lines because they cross meridians at 90° angles. Additionally, all meridians are rhumb lines.

In general, rhumb lines spiral toward one of the poles. If the azimuth of a rhumb line is true east, west, north, or south, then the rhumb line connects with itself to form a small circle or a pair of antipodal meridians.

Rhumb lines are useful for navigation because the bearing (azimuth) does not change along the route. While rhumb line paths are longer than great circle paths, the constant bearing makes rhumb line paths easier to navigate.

Great Circles

A great circle is the shortest path between two points along the surface of a sphere, as defined by the intersection of the surface of the sphere and a plane passing through the center of the sphere. Great circles always bisect the sphere.

The equator and all meridians are both great circles and rhumb lines. Other great circles are not rhumb lines because they do not have a constant azimuth. Instead, they cross successive meridians at different angles.

Great circles are examples of geodesics. A geodesic is the shortest path between two points on a curved surface, such as the shortest path on an ellipsoid.

Great circles are less useful for navigation because, in general, the bearing changes along the route. While great circle paths are shorter than rhumb line paths, the changes in bearing make great circle paths more difficult to navigate.

Use Rhumb Lines and Great Circles in Functions

Several geometric geodesy functions enable you to specify whether a path follows a rhumb line or a great circle. For example, you can generate track points along either a rhumb line or great circle path by using the track2 function. You can also find the length of the rhumb line or great circle path that connects two points by using the distance function.