Great ellipse

For the shortest path between two points on the earth, see geodesics on an ellipsoid.

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.

A spheroid

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—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 ):

• 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

  ${\begin{aligned}\tan \alpha &={\frac {\tan \gamma }{\sqrt {1-e^{2}\cos ^{2}\beta }}},\\\tan \gamma &={\frac {\tan \alpha }{\sqrt {1+e'^{2}\cos ^{2}\phi }}},\end{aligned}}$

  and the quadrants of $\alpha$ and $\gamma$ are the same.
• 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.

See also

• Earth section paths
• Great-circle navigation
• Geodesics on an ellipsoid
• Meridian arc
• Rhumb line

External links

• Matlab implementation of the solutions for the direct and inverse problems for great ellipses.