Great ellipse

A great ellipse is an ellipse passing through two points on a spheroid and having the same center as that of the spheroid.

Equivalently, it is an ellipse on the surface of a spheroid and centered on the origin, or the curve formed by intersecting the spheroid by a plane through its center.

[1] For points that are separated by less than about a quarter of the circumference of the earth, about

, the length of the great ellipse connecting the points is close (within one part in 500,000) to the geodesic distance.

[2][3][4] The great ellipse therefore is sometimes proposed as a suitable route for marine navigation.

The great ellipse is special case of an earth section path.

Assume that the spheroid, an ellipsoid of revolution, has an equatorial radius

There are various ways to map an ellipsoid into a sphere of radius

in such a way as to map the great ellipse into a great circle, allowing the methods of great-circle navigation to be used: The last method gives an easy way to generate a succession of way-points on the great ellipse connecting two known points

Solve for the great circle between

and find the way-points on the great circle.

These map into way-points on the corresponding great ellipse.

If distances and headings are needed, it is simplest to use the first of the mappings.

[5] In detail, the mapping is as follows (this description is taken from [6]):

(A similar mapping to an auxiliary sphere is carried out in the solution of geodesics on an ellipsoid.

is conserved in the mapping, while the longitude

maps to a "spherical" longitude

The equivalent ellipse used for distance calculations has semi-axes

The "inverse problem" is the determination of

The spherical azimuths are relabeled as

and the spherical azimuths at the equator and at

The azimuths of the endpoints of great ellipse,

The semi-axes of the great ellipse can be found using the value of

Also determined as part of the solution of the great circle problem are the arc lengths,

, measured from the equator crossing to

is found by computing the length of a portion of perimeter of the ellipse using the formula giving the meridian arc in terms the parametric latitude.

In applying this formula, use the semi-axes for the great ellipse (instead of for the meridian) and substitute

The solution of the "direct problem", determining the position of

, can be similarly be found (this requires, in addition, the inverse meridian distance formula).

This also enables way-points (e.g., a series of equally spaced intermediate points) to be found in the solution of the inverse problem.