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. For points that are separated by less than about a quarter of the circumference of the earth, about $$10\,000\,\mathrm{km}$$, 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. The great ellipse is special case of an earth section path.

Introduction
Assume that the spheroid, an ellipsoid of revolution, has an equatorial radius $$a$$ and polar semi-axis $$b$$. Define the flattening $$f=(a-b)/a$$, the eccentricity $$e=\sqrt{f(2-f)}$$, and the second eccentricity $$e'=e/(1-f)$$. Consider two points: $$A$$ at (geographic) latitude $$\phi_1$$ and longitude $$\lambda_1$$ and $$B$$ at latitude $$\phi_2$$ and longitude $$\lambda_2$$. The connecting great ellipse (from $$A$$ to $$B$$) has length $$s_{12}$$ and has azimuths $$\alpha_1$$ and $$\alpha_2$$ at the two endpoints.

There are various ways to map an ellipsoid into a sphere of radius $$a$$ 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 ellipsoid can be stretched in a direction parallel to the axis of rotation; this maps a point of latitude $$\phi$$ on the ellipsoid to a point on the sphere with latitude $$\beta$$, the parametric latitude.
 * A point on the ellipsoid can mapped radially onto the sphere along the line connecting it with the center of the ellipsoid; this maps a point of latitude $$\phi$$ on the ellipsoid to a point on the sphere with latitude $$\theta$$, the geocentric latitude.
 * The ellipsoid can be stretched into a prolate ellipsoid with polar semi-axis $$a^2/b$$ and then mapped radially onto the sphere; this preserves the latitude&mdash;the latitude on the sphere is $$\phi$$, the geographic latitude.

The last method gives an easy way to generate a succession of way-points on the great ellipse connecting two known points $$A$$ and $$B$$. Solve for the great circle between $$(\phi_1,\lambda_1)$$ and $$(\phi_2,\lambda_2)$$ and find the way-points on the great circle. These map into way-points on the corresponding great ellipse.

Mapping the great ellipse to a great circle
If distances and headings are needed, it is simplest to use the first of the mappings. In detail, the mapping is as follows (this description is taken from ):

\begin{align} \tan\alpha &= \frac{\tan\gamma}{\sqrt{1-e^2\cos^2\beta}}, \\ \tan\gamma &= \frac{\tan\alpha}{\sqrt{1+e'^2\cos^2\phi}}, \end{align} $$ and the quadrants of $$\alpha$$ and $$\gamma$$ are the same.
 * The geographic latitude $$\phi$$ on the ellipsoid maps to the parametric latitude $$\beta$$ on the sphere, where"$a\tan\beta = b\tan\phi.$"
 * The longitude $$\lambda$$ is unchanged.
 * The azimuth $$\alpha$$ on the ellipsoid maps to an azimuth $$\gamma$$ on the sphere where $$
 * Positions on the great circle of radius $$a$$ are parametrized by arc length $$\sigma$$ measured from the northward crossing of the equator. The great ellipse has a semi-axes $$a$$ and $$a \sqrt{1 - e^2\cos^2\gamma_0}$$, where $$\gamma_0$$ is the great-circle azimuth at the northward equator crossing, and $$\sigma$$ is the parametric angle on the ellipse.

(A similar mapping to an auxiliary sphere is carried out in the solution of geodesics on an ellipsoid. The differences are that the azimuth $$\alpha$$ is conserved in the mapping, while the longitude $$\lambda$$ maps to a "spherical" longitude $$\omega$$. The equivalent ellipse used for distance calculations has semi-axes $$b \sqrt{1 + e'^2\cos^2\alpha_0}$$ and $$b$$.)

Solving the inverse problem
The "inverse problem" is the determination of $$s_{12}$$, $$\alpha_1$$, and $$\alpha_2$$, given the positions of $$A$$ and $$B$$. This is solved by computing $$\beta_1$$ and $$\beta_2$$ and solving for the great-circle between $$(\beta_1,\lambda_1)$$ and $$(\beta_2,\lambda_2)$$.

The spherical azimuths are relabeled as $$\gamma$$ (from $$\alpha$$). Thus $$\gamma_0$$, $$\gamma_1$$, and $$\gamma_2$$ and the spherical azimuths at the equator and at $$A$$ and $$B$$. The azimuths of the endpoints of great ellipse, $$\alpha_1$$ and $$\alpha_2$$, are computed from $$\gamma_1$$ and $$\gamma_2$$.

The semi-axes of the great ellipse can be found using the value of $$\gamma_0$$.

Also determined as part of the solution of the great circle problem are the arc lengths, $$\sigma_{01}$$ and $$\sigma_{02}$$, measured from the equator crossing to $$A$$ and $$B$$. The distance $$s_{12}$$ 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 $$\sigma_{01}$$ and $$\sigma_{02}$$ for $$\beta$$.

The solution of the "direct problem", determining the position of $$B$$ given $$A$$, $$\alpha_1$$, and $$s_{12}$$, 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.