Great ellipse

Short description: Ellipse on a spheroid centered on its origin

right|150px|thumb|A spheroid

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 [math]\displaystyle{ 10\,000\,\mathrm{km} }[/math], 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 [math]\displaystyle{ a }[/math] and polar semi-axis [math]\displaystyle{ b }[/math]. Define the flattening [math]\displaystyle{ f=(a-b)/a }[/math], the eccentricity [math]\displaystyle{ e=\sqrt{f(2-f)} }[/math], and the second eccentricity [math]\displaystyle{ e'=e/(1-f) }[/math]. Consider two points: [math]\displaystyle{ A }[/math] at (geographic) latitude [math]\displaystyle{ \phi_1 }[/math] and longitude [math]\displaystyle{ \lambda_1 }[/math] and [math]\displaystyle{ B }[/math] at latitude [math]\displaystyle{ \phi_2 }[/math] and longitude [math]\displaystyle{ \lambda_2 }[/math]. The connecting great ellipse (from [math]\displaystyle{ A }[/math] to [math]\displaystyle{ B }[/math]) has length [math]\displaystyle{ s_{12} }[/math] and has azimuths [math]\displaystyle{ \alpha_1 }[/math] and [math]\displaystyle{ \alpha_2 }[/math] at the two endpoints.

There are various ways to map an ellipsoid into a sphere of radius [math]\displaystyle{ a }[/math] 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 [math]\displaystyle{ \phi }[/math] on the ellipsoid to a point on the sphere with latitude [math]\displaystyle{ \beta }[/math], 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 [math]\displaystyle{ \phi }[/math] on the ellipsoid to a point on the sphere with latitude [math]\displaystyle{ \theta }[/math], the geocentric latitude.
• The ellipsoid can be stretched into a prolate ellipsoid with polar semi-axis [math]\displaystyle{ a^2/b }[/math] and then mapped radially onto the sphere; this preserves the latitude—the latitude on the sphere is [math]\displaystyle{ \phi }[/math], 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 [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math]. Solve for the great circle between [math]\displaystyle{ (\phi_1,\lambda_1) }[/math] and [math]\displaystyle{ (\phi_2,\lambda_2) }[/math] 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 [math]\displaystyle{ \phi }[/math] on the ellipsoid maps to the parametric latitude [math]\displaystyle{ \beta }[/math] on the sphere, where

  [math]\displaystyle{ a\tan\beta = b\tan\phi. }[/math]

• The longitude [math]\displaystyle{ \lambda }[/math] is unchanged.
• The azimuth [math]\displaystyle{ \alpha }[/math] on the ellipsoid maps to an azimuth [math]\displaystyle{ \gamma }[/math] on the sphere where

  [math]\displaystyle{ \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} }[/math]

  and the quadrants of [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \gamma }[/math] are the same.
• Positions on the great circle of radius [math]\displaystyle{ a }[/math] are parametrized by arc length [math]\displaystyle{ \sigma }[/math] measured from the northward crossing of the equator. The great ellipse has a semi-axes [math]\displaystyle{ a }[/math] and [math]\displaystyle{ a \sqrt{1 - e^2\cos^2\gamma_0} }[/math], where [math]\displaystyle{ \gamma_0 }[/math] is the great-circle azimuth at the northward equator crossing, and [math]\displaystyle{ \sigma }[/math] 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 [math]\displaystyle{ \alpha }[/math] is conserved in the mapping, while the longitude [math]\displaystyle{ \lambda }[/math] maps to a "spherical" longitude [math]\displaystyle{ \omega }[/math]. The equivalent ellipse used for distance calculations has semi-axes [math]\displaystyle{ b \sqrt{1 + e'^2\cos^2\alpha_0} }[/math] and [math]\displaystyle{ b }[/math].)

Solving the inverse problem

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

The spherical azimuths are relabeled as [math]\displaystyle{ \gamma }[/math] (from [math]\displaystyle{ \alpha }[/math]). Thus [math]\displaystyle{ \gamma_0 }[/math], [math]\displaystyle{ \gamma_1 }[/math], and [math]\displaystyle{ \gamma_2 }[/math] and the spherical azimuths at the equator and at [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math]. The azimuths of the endpoints of great ellipse, [math]\displaystyle{ \alpha_1 }[/math] and [math]\displaystyle{ \alpha_2 }[/math], are computed from [math]\displaystyle{ \gamma_1 }[/math] and [math]\displaystyle{ \gamma_2 }[/math].

The semi-axes of the great ellipse can be found using the value of [math]\displaystyle{ \gamma_0 }[/math].

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

The solution of the "direct problem", determining the position of [math]\displaystyle{ B }[/math] given [math]\displaystyle{ A }[/math], [math]\displaystyle{ \alpha_1 }[/math], and [math]\displaystyle{ s_{12} }[/math], 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.