Английская Википедия:Great ellipse
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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 <math>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.[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.
Introduction
Assume that the spheroid, an ellipsoid of revolution, has an equatorial radius <math>a</math> and polar semi-axis <math>b</math>. Define the flattening <math>f=(a-b)/a</math>, the eccentricity <math>e=\sqrt{f(2-f)}</math>, and the second eccentricity <math>e'=e/(1-f)</math>. Consider two points: <math>A</math> at (geographic) latitude <math>\phi_1</math> and longitude <math>\lambda_1</math> and <math>B</math> at latitude <math>\phi_2</math> and longitude <math>\lambda_2</math>. The connecting great ellipse (from <math>A</math> to <math>B</math>) has length <math>s_{12}</math> and has azimuths <math>\alpha_1</math> and <math>\alpha_2</math> at the two endpoints.
There are various ways to map an ellipsoid into a sphere of radius <math>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>\phi</math> on the ellipsoid to a point on the sphere with latitude <math>\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>\phi</math> on the ellipsoid to a point on the sphere with latitude <math>\theta</math>, the geocentric latitude.
- The ellipsoid can be stretched into a prolate ellipsoid with polar semi-axis <math>a^2/b</math> and then mapped radially onto the sphere; this preserves the latitude—the latitude on the sphere is <math>\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>A</math> and <math>B</math>. Solve for the great circle between <math>(\phi_1,\lambda_1)</math> and <math>(\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.[5] In detail, the mapping is as follows (this description is taken from [6]):
- The geographic latitude <math>\phi</math> on the ellipsoid maps to the parametric latitude <math>\beta</math> on the sphere, where
<math>a\tan\beta = b\tan\phi.</math>
- The longitude <math>\lambda</math> is unchanged.
- The azimuth <math>\alpha</math> on the ellipsoid maps to an azimuth <math>\gamma</math> on the sphere where
<math>
\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>\alpha</math> and <math>\gamma</math> are the same.
- Positions on the great circle of radius <math>a</math> are parametrized by arc length <math>\sigma</math> measured from the northward crossing of the equator. The great ellipse has a semi-axes <math>a</math> and <math>a \sqrt{1 - e^2\cos^2\gamma_0}</math>, where <math>\gamma_0</math> is the great-circle azimuth at the northward equator crossing, and <math>\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>\alpha</math> is conserved in the mapping, while the longitude <math>\lambda</math> maps to a "spherical" longitude <math>\omega</math>. The equivalent ellipse used for distance calculations has semi-axes <math>b \sqrt{1 + e'^2\cos^2\alpha_0}</math> and <math>b</math>.)
Solving the inverse problem
The "inverse problem" is the determination of <math>s_{12}</math>, <math>\alpha_1</math>, and <math>\alpha_2</math>, given the positions of <math>A</math> and <math>B</math>. This is solved by computing <math>\beta_1</math> and <math>\beta_2</math> and solving for the great-circle between <math>(\beta_1,\lambda_1)</math> and <math>(\beta_2,\lambda_2)</math>.
The spherical azimuths are relabeled as <math>\gamma</math> (from <math>\alpha</math>). Thus <math>\gamma_0</math>, <math>\gamma_1</math>, and <math>\gamma_2</math> and the spherical azimuths at the equator and at <math>A</math> and <math>B</math>. The azimuths of the endpoints of great ellipse, <math>\alpha_1</math> and <math>\alpha_2</math>, are computed from <math>\gamma_1</math> and <math>\gamma_2</math>.
The semi-axes of the great ellipse can be found using the value of <math>\gamma_0</math>.
Also determined as part of the solution of the great circle problem are the arc lengths, <math>\sigma_{01}</math> and <math>\sigma_{02}</math>, measured from the equator crossing to <math>A</math> and <math>B</math>. The distance <math>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>\sigma_{01}</math> and <math>\sigma_{02}</math> for <math>\beta</math>.
The solution of the "direct problem", determining the position of <math>B</math> given <math>A</math>, <math>\alpha_1</math>, and <math>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
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