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According to Wikipedia:

The great-circle distance or orthodromic distance is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a straight line through the sphere's interior). The distance between two points in Euclidean space is the length of a straight line between them, but on the sphere there are no straight lines. In spaces with curvature, straight lines are replaced by geodesics. Geodesics on the sphere are circles on the sphere whose centers coincide with the center of the sphere, and are called great circles.

Note that the definition also mentions 'Geodesics'

Operationally what is the difference between them?

In ArcMap, the measurement tool allows measuring 'Geodesic' distances but not 'Great Circle' distances. Can these be considered equivalent?

A similar question regarding Planar vs Euclidean distance can be found at Difference between planar and Euclidean distance?

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Geodesics are the shortest path on a curved surface (e.g. sphere or ellipsoid) - like a straight line on a flat plane. A great circle is the shortest path on a sphere, but not on an ellipsoid (as long as both axes are not of equal length). So every great circle is a geodesic, but not every geodesic must be a great circle (on an ellipsoid for example)

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Geodesic curve: Curve which geodesic curvature is null

Geodesics are a family of curves of zero geodesic curvature.

  • On a plane, the geodesic between two points is a line.
  • On a sphere it is an arc of great circle.
  • On an ellipsoid, it is close to a circle, but generally an open curve: enter image description here
    (Source)
  • On complex surfaces, geodesics are a mix of various shapes, including lines:

enter image description here

(Red, blue, green curves are all geodesics, source)

Geodesic distance on ellipsoid: Vicenty's formula

A geodesic distance is one measured along a geodesic. On a sphere it is measured along a great circle.

On an ellipsoid with a small eccentricity the geodesic is close to a great circle. The length of this geodesic is usually approximated iteratively using Vicenty's formula.

Geodetic, geodesic: Synonymous

There is no reason for the adjective geodetic to be used to qualify a geodesic curve. The origin seems to be an imprecise translation from French. However the confusion is present and in practical both geodesic and geodetic are found. WGS 84 uses geodetic, so I guess the future is to geodetic.

Geodesic, shortest path: Not synonymous

On any surface, a person walking "straight ahead" (or a small vehicle with un-steerable wheels) follows a geodesic.

The shortest path on a given surface can be a geodesic. However this is misleading:

  • There are geodesics which are not the shortest paths: The shortest path on a sphere is an arc of great circle, but the complement of this arc, which generally is not the shortest path, is also a geodesic.

  • There are shortest paths which are not geodesics. The black geodesic is not the shortest path from A to B:

    enter image description here
    Source.

No geodesic curvature = no torsion

As a geodesic have no geodesic curvature, it can be used to remove torsion forces on stiffeners. This led to geodesic airframes and ship hulls.

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