Great circle: definition, properties, history and applications
A great circle is the largest circle on a sphere; it divides the sphere into equal hemispheres, is a geodesic, and gives the shortest surface path between points—used in navigation, mapping and geometry.
Overview
A great circle is a circle on the surface of a sphere whose center coincides with the center of the sphere; equivalently, it is the intersection of the sphere with a plane that passes through the sphere's center. As the largest possible circle that can be drawn on a sphere, a great circle divides the surface into two equal halves called hemispheres and has the same radius and circumference as the generating sphere. In spherical geometry a great circle plays the role analogous to a straight line in plane geometry and represents a locally shortest route, or geodesic, between points.
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Mathematical definition and properties
Formally, a great circle is obtained when a sphere is cut by a plane through its center. Every great circle has the same circumference and is congruent to every other great circle on the same sphere. Given two distinct, non-antipodal points on a sphere there is exactly one great circle passing through them; if the two points are antipodal (directly opposite) there are infinitely many great circles through them. Great circles are geodesics on the sphere and are characterized by having zero geodesic curvature.
- Intersection description: sphere ∩ plane through center (definition).
- Hemispheres: divides the sphere into equal halves (hemisphere).
- Geodesic property: shortest surface path (geodesic).
- Antipodal exception: infinite great circles through opposite points (antipodes).
Examples and familiar great circles
On Earth the most familiar great circles are the Equator and the meridians (longitudes). The Equator is a great circle that lies in the plane perpendicular to the Earth's rotational axis and divides Earth into Northern and Southern Hemispheres. Each line of longitude, when extended into a full circle joining the North and South Poles, is also a great circle. Other great circles may be oriented arbitrarily, and there are infinitely many on an ideal sphere (example, meridians, equator).
History and practical uses
Ancient navigators and astronomers understood that long-distance routes on a curved Earth should follow arcs close to great circles to minimize travel. In modern practice, airlines and shipping often plan routes along great-circle paths to reduce distance and fuel consumption. On flat maps a great-circle route typically appears curved (except on some projections), which is why map projections such as the Mercator distort the appearance of shortest routes (navigation, aviation, mapping).
Computation and distinctions
Computing a great-circle distance between two points on a sphere uses spherical trigonometry or formulae such as the haversine or the spherical law of cosines. For practical Earth-distance work, the oblate shape of the planet is sometimes taken into account, but the great-circle model is a useful first approximation. Distinctions to note: a rhumb line (loxodrome) crosses meridians at constant bearing and is not generally shortest, while a great circle does not preserve compass heading except at the start and end of a segment (haversine, spherical law of cosines, rhumb line).
Notable facts and extensions
Great circles appear in multiple fields: in cartography and navigation, in geometry and topology as examples of closed geodesics, and in engineering where spherical intersections are relevant (for instance, geodesic domes use concepts related to great-circle arcs). In higher mathematics the concept generalizes to great spheres and great hyperspheres on n-spheres. For additional technical background and examples see reference 14, reference 15, and reference 16. Further reading and resources: resource 17, resource 18, resource 19.
Summary: A great circle is the intersection of a sphere and a plane through its center, is the largest circle on that sphere, constitutes geodesics that give shortest surface paths, and is central to navigation, mapping and spherical geometry.
Display on maps
Since many maps (e.g. the Mercator map) are drawn in such a way that the latitudes appear as straight, horizontal lines, the flight paths appear curved despite their shortness and run further polewards (see also rhumb lines). To simplify drawing, there are special great circle maps (see gnomonic projection) on which all great circles appear as straight lines, but the surrounding area is somewhat distorted. Shortest air routes and long shortest ship routes (e.g. when crossing the Atlantic) can be shown as straight lines on a gnomonic map. The compass course to be followed changes continuously and can be read on the map as the angle between the meridian and the course line.
On nautical charts, the latitude is plotted on the right and left edges, i.e. the respective section of the relevant longitude great circle. Here the navigator can take a distance with the compass (1 minute of arc = 1 nautical mile = 1.852 km) and transfer it to the chart to plot positions and courses.
Calculation
The angle between points A and B with latitude coordinates φ 
and longitude coordinates λ 
on the great circle is calculated as follows:
If 
is given in radians, the great circle distance d between the two points can be calculated from the earth radius rE:
The great circle distance is at most half the circumference of the earth.
The angle of intersection of the great circle of A and B with the meridian at point A is called the heading angle ω 
. It is calculated by:
For easterly courses (λB > λA) the course angle is between 0° and 180°, for westerly courses (λB < λA) the course angle is between 180° and 360°. In contrast to plane geometry, the course angles from A to B and from B to A do not differ by 180°. In the extreme case, when the great circle passes over the poles, the two course angles can even be equal.
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AlegsaOnline.com Great circle: definition, properties, history and applications Leandro Alegsa
URL: https://en.alegsaonline.com/art/40480