150px, A ">spheroid
A great ellipse is an
ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
passing through two
points on a
spheroid
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has cir ...
and having the same
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
as that of the spheroid. Equivalently, it is an ellipse on the
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
of a spheroid and centered on the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
, 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
Earth's circumference is the distance around Earth. Measured around the Equator, it is . Measured around the poles, the circumference is .
Measurement of Earth's circumference has been important to navigation since ancient times. The first kno ...
, about
, the length of the great ellipse connecting the points is close (within one part in 500,000) to the
geodesic distance
In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them. This is also known as the geodesic distance or shortest-path dista ...
.
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
and polar semi-axis
. Define the flattening
, the eccentricity
, and the second eccentricity
. Consider two points:
at (geographic) latitude
and longitude
and
at latitude
and longitude
. The connecting great ellipse (from
to
) has length
and has
azimuth
An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north.
Mathematicall ...
s
and
at the two endpoints.
There are various ways to map an ellipsoid into a sphere of radius
in such a way as to map the great ellipse into a great circle, allowing the methods of
great-circle navigation
Great-circle navigation or orthodromic navigation (related to orthodromic course; from the Greek ''ορθóς'', right angle, and ''δρóμος'', path) is the practice of navigating a vessel (a ship or aircraft) along a great circle. Such rout ...
to be used:
* The ellipsoid can be stretched in a direction parallel to the axis of rotation; this maps a point of latitude
on the ellipsoid to a point on the sphere with latitude
, the
parametric latitude
In geography, latitude is a coordinate that specifies the north–south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pole ...
.
* 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
on the ellipsoid to a point on the sphere with latitude
, the
geocentric latitude
In geography, latitude is a coordinate that specifies the north–south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pole ...
.
* The ellipsoid can be stretched into a prolate ellipsoid with polar semi-axis
and then mapped radially onto the sphere; this preserves the latitude—the latitude on the sphere is
, the
geographic latitude
In geography, latitude is a coordinate that specifies the north–south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pole ...
.
The last method gives an easy way to generate a succession of way-points on the great ellipse connecting two known points
and
. Solve for the great circle between
and
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
on the ellipsoid maps to the parametric latitude
on the sphere, where
* The longitude
is unchanged.
* The azimuth
on the ellipsoid maps to an azimuth
on the sphere where
and the quadrants of
and
are the same.
* Positions on the great circle of radius
are parametrized by arc length
measured from the northward crossing of the equator. The great ellipse has a semi-axes
and
, where
is the great-circle azimuth at the northward equator crossing, and
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 study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an ''oblate ellipsoid'', a slightly flattened sphere. A ''geodes ...
. The differences are that the azimuth
is conserved in the mapping, while the longitude
maps to a "spherical" longitude
. The equivalent ellipse used for distance calculations has semi-axes
and
.)
Solving the inverse problem
The "inverse problem" is the determination of
,
, and
, given the positions of
and
. This is solved by computing
and
and solving for the
great-circle between
and
.
The spherical azimuths are relabeled as
(from
). Thus
,
, and
and the spherical azimuths at the equator and at
and
. The azimuths of the endpoints of great ellipse,
and
, are computed from
and
.
The semi-axes of the great ellipse can be found using the value of
.
Also determined as part of the solution of the great circle problem are the arc lengths,
and
, measured from the equator crossing to
and
. The distance
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
and
for
.
The solution of the "direct problem", determining the position of
given
,
, and
, 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
Great-circle navigation or orthodromic navigation (related to orthodromic course; from the Greek ''ορθóς'', right angle, and ''δρóμος'', path) is the practice of navigating a vessel (a ship or aircraft) along a great circle. Such rout ...
*
Geodesics on an ellipsoid
The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an ''oblate ellipsoid'', a slightly flattened sphere. A ''geodes ...
*
Meridian arc
In geodesy and navigation, a meridian arc is the curve between two points on the Earth's surface having the same longitude. The term may refer either to a segment of the meridian, or to its length.
The purpose of measuring meridian arcs is to de ...
*
Rhumb line
In navigation, a rhumb line, rhumb (), or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true north.
Introduction
The effect of following a rhumb li ...
References
{{reflist
External links
Matlab implementation of the solutions for the direct and inverse problems for great ellipses.
Geometry
Curves