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 spheroid and centered at the origin, or the curve formed by intersecting the spheroid by a plane through its center. For points which are separated by less than about a quarter of the circumference of the earth, about ${\displaystyle 10\,000\,\mathrm {km} }$, the length of the great ellipse connecting the points is close (within one part in 500,000) to the geodesic distance. The great ellipse therefore is sometimes proposed as a suitable route for marine navigation.

Assume that the spheroid, an ellipsoid of revolution, has an equatorial radius ${\displaystyle a}$ and polar semi-axis ${\displaystyle b}$. Define the flattening ${\displaystyle f=(a-b)/a}$, the eccentricity ${\displaystyle e={\sqrt {f(2-f)}}}$, and the second eccentricity ${\displaystyle e'=e/(1-f)}$. Consider two points: ${\displaystyle A}$ at (geographic) latitude ${\displaystyle \phi _{1}}$ and longitude ${\displaystyle \lambda _{1}}$ and ${\displaystyle B}$ at latitude ${\displaystyle \phi _{2}}$ and longitude ${\displaystyle \lambda _{2}}$. The connecting great ellipse (from ${\displaystyle A}$ to ${\displaystyle B}$) has length ${\displaystyle s_{12}}$ and has azimuths ${\displaystyle \alpha _{1}}$ and ${\displaystyle \alpha _{2}}$ at the two endpoints.

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