An ellipsoid is a quadric surface; that is, a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, an ellipsoid is characterized by either of the two following properties. Every planar cross section is either an ellipse, or is empty, or is reduced to a single point (this explains the name, meaning "ellipse like"). It is bounded, which means that it may be enclosed in a sufficiently large sphere.
An ellipsoid has three pairwise perpendicular axes of symmetry which intersect at a center of symmetry, called the center of the ellipsoid. The line segments that are delimited on the axes of symmetry by the ellipsoid are called the principal axes, or simply axes of the ellipsoid. If the three axes have different lengths, the ellipsoid is said to be tri-axial or (rarely) scalene, and the axes are uniquely defined.
If two of the axes have the same length, then the ellipsoid is an "ellipsoid of revolution", also called a spheroid. In this case, the ellipsoid is invariant under a rotation around the third axis, and there are thus infinitely many ways of choosing the two perpendicular axes of the same length. If the third axis is shorter, the ellipsoid is an oblate spheroid; if it is longer, it is a prolate spheroid. If the three axes have the same length, the ellipsoid is a sphere.
where a, b, c are positive real numbers.
The points (a, 0, 0), (0, b, 0) and (0, 0, c) lie on the surface. The line segments from the origin to these points are called the principal semi-axes of the ellipsoid, because a, b, c are half the length of the principal axes. They correspond to the semi-major axis and semi-minor axis of an ellipse.
The ellipsoid may be parameterized in several ways, which are simpler to express when the ellipsoid axes coincide with coordinate axes. A common choice is
These parameters may be interpreted as spherical coordinates, where is the polar angle, and is the azimuth angle of the point (x, y, z) of the ellipsoid.
The volume bounded by the ellipsoid is
which, as follows from basic trigonometric identities, are equivalent expressions (i.e. the formula for can be used to calculate the surface area of a prolate ellipsoid and vice versa). In both cases e may again be identified as the eccentricity of the ellipse formed by the cross section through the symmetry axis. (See ellipse). Derivations of these results may be found in standard sources, for example Mathworld.
Here p ≈ 1.6075 yields a relative error of at most 1.061%; a value of p = 8/5 = 1.6 is optimal for nearly spherical ellipsoids, with a relative error of at most 1.178%.
In the "flat" limit of c much smaller than a, b, the area is approximately 2πab, equivalent to p ≈ 1.5850.
The intersection of a plane and a sphere is a circle (or is reduced to a single point, or is empty). Any ellipsoid is the image of the unit sphere under some affine transformation, and any plane is the image of some other plane under the same transformation. So, because affine transformations map circles to ellipses, the intersection of a plane with an ellipsoid is an ellipse or a single point, or is empty. Obviously, spheroids contain circles. This is also true, but less obvious, for triaxial ellipsoids (see Circular section).
Given: Ellipsoid and the plane with equation which have an ellipse in common.
Wanted: Three vectors (center) and (conjugate vectors), such that the ellipse can be represented by the parametric equation
Solution: The scaling transforms the ellipsoid onto the unit sphere and the given plane onto the plane with equation . Let be the Hesse normal form of the new plane and its unit normal vector. Hence
is the center of the intersection circle and its radius (See diagram).
In case of let be (The plane is horizontal !)
In case of let be
In any case the vectors are orthogonal, parallel to the intersection plane and have length (radius of the circle). Hence the intersection circle can be described by the parametric equation
The reverse scaling (See above) transforms the unit sphere back to the ellipsoid and the vectors are mapped onto vectors , which were wanted for the parametric representation of the intersection ellipse.
How to find the vertices and semi-axes of the ellipse is described in ellipse.
Example: The diagrams show an ellipsoid with the semi-axes which is cut by the plane
More generally, an arbitrarily oriented ellipsoid, centered at v, is defined by the solutions x to the equation
The eigenvectors of A define the principal axes of the ellipsoid and the eigenvalues of A are the reciprocals of the squares of the semi-axes: , and . An invertible linear transformation applied to a sphere produces an ellipsoid, which can be brought into the above standard form by a suitable rotation, a consequence of the polar decomposition (also, see spectral theorem). If the linear transformation is represented by a symmetric 3-by-3 matrix, then the eigenvectors of the matrix are orthogonal (due to the spectral theorem) and represent the directions of the axes of the ellipsoid; the lengths of the semi-axes are computed from the eigenvalues. The singular value decomposition and polar decomposition are matrix decompositions closely related to these geometric observations.
The key to a parametric representation of an ellipsoid in general position is the alternative definition:
An affine transformation can be represented by a translation with a vector and a regular 3×3-matrix :
where are the column vectors of matrix .
A parametric representation of an ellipsoid in general position can be obtained by the parametric representation of a unit sphere (see above) and an affine transformation:
If the vectors form an orthogonal system, the points with vectors are the vertices of the ellipsoid and are the semi principal axes.
A surface normal vector at point is
The ellipsoidal shape finds many practical applications:
The mass of an ellipsoid of uniform density ρ is:
The moments of inertia of an ellipsoid of uniform density are:
For these moments of inertia reduce to those for a sphere of uniform density.
Ellipsoids and cuboids rotate stably along their major or minor axes, but not along their median axis. This can be seen experimentally by throwing an eraser with some spin. In addition, moment of inertia considerations mean that rotation along the major axis is more easily perturbed than rotation along the minor axis.
One practical effect of this is that scalene astronomical bodies such as Haumea generally rotate along their minor axes (as does Earth, which is merely oblate); in addition, because of tidal locking, moons in synchronous orbit such as Mimas orbit with their major axis aligned radially to their planet.
A spinning body of homogeneous self-gravitating fluid will assume the form of either a Maclaurin spheroid (oblate spheroid) or Jacobi ellipsoid (scalene ellipsoid) when in hydrostatic equilibrium, and for moderate rates of rotation. At faster rotations, non-ellipsoidal piriform or oviform shapes can be expected, but these are not stable.
The ellipsoid is the most general shape for which it has been possible to calculate the creeping flow of fluid around the solid shape. The calculations include the force required to translate through a fluid and to rotate within it. Applications include determining the size and shape of large molecules, the sinking rate of small particles, and the swimming abilities of microorganisms.
The elliptical distributions, which generalize the multivariate normal distribution and are used in finance, can be defined in terms of their density functions. When they exist, the density functions f have the structure:
where is a scale factor, is an -dimensional random row vector with median vector (which is also the mean vector if the latter exists), is a positive definite matrix which is proportional to the covariance matrix if the latter exists, and is a function mapping from the non-negative reals to the non-negative reals giving a finite area under the curve. The multivariate normal distribution is the special case in which for quadratic form .
Thus the density function is a scalar-to-scalar transformation of a quadric expression. Moreover, the equation for any iso-density surface states that the quadric expression equals some constant specific to that value of the density, and the iso-density surface is an ellipsoid.
The volume of a higher-dimensional ellipsoid (a hyperellipsoid) can be calculated using the dimensional constant given for the volume of a hypersphere. One can also define hyperellipsoids as the images of spheres under invertible linear transformations. The spectral theorem can again be used to obtain a standard equation akin to the one given above.
Diffusion-weighted magnetic resonance imaging (DWI or DW-MRI) is the use of specific MRI sequences as well as software that generates images from the resulting data, that uses the diffusion of water molecules to generate contrast in MR images. It allows the mapping of the diffusion process of molecules, mainly water, in biological tissues, in vivo and non-invasively. Molecular diffusion in tissues is not free, but reflects interactions with many obstacles, such as macromolecules, fibers, and membranes. Water molecule diffusion patterns can therefore reveal microscopic details about tissue architecture, either normal or in a diseased state. A special kind of DWI, diffusion tensor imaging (DTI), has been used extensively to map white matter tractography in the brain.Earth ellipsoid
An Earth ellipsoid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy, and the geosciences. Various different ellipsoids have been used as approximations.
It is an ellipsoid of revolution whose minor axis (shorter diameter), which connects the geographical North Pole and South Pole, is approximately aligned with the Earth's axis of rotation. The ellipsoid is defined by the equatorial axis a and the polar axis b; their difference is about 21 km, or 0.335%. Additional parameters are the mass function J2, the correspondent gravity formula, and the rotation period (usually 86164 seconds).
Many methods exist for determination of the axes of an Earth ellipsoid, ranging from meridian arcs up to modern satellite geodesy or the analysis and interconnection of continental geodetic networks. Amongst the different set of data used in national surveys are several of special importance: the Bessel ellipsoid of 1841, the international Hayford ellipsoid of 1924, and (for GPS positioning) the WGS84 ellipsoid.Earth radius
Earth radius is the distance from a selected center of Earth to a point on its surface, which is often chosen to be sea level, or more commonly, the surface of an idealized ellipsoid representing the shape of Earth. Because Earth is not a perfect sphere, the determination of Earth's radius can have several values, depending on how it is measured; from its equatorial radius of about 6,378 kilometres (3,963 miles) to its polar radius of about 6,357 kilometres (3,950 miles).
When only one radius is stated, the International Astronomical Union (IAU) prefers that it be Earth's equatorial radius.The International Union of Geodesy and Geophysics (IUGG) gives three global average radii, the arithmetic mean of the radii of the ellipsoid (R1), the radius of a sphere with the same surface area as the ellipsoid or authalic radius (R2), and the radius of a sphere with the same volume as the ellipsoid (R3). All three IUGG average radii are about 6,371 kilometres (3,959 mi). A fourth global average radius not mentioned by the IUGG is the rectifying radius, the radius of a sphere with a circumference equal to the perimeter of the polar cross section of the ellipsoid, about 6,367 kilometres (3,956 mi). The radius of curvature at any point on the surface of the ellipsoid depends on its coordinates and its azimuth, north-south (meridional), east-west (prime vertical), or somewhere in between.Ellipsoid method
In mathematical optimization, the ellipsoid method is an iterative method for minimizing convex functions. When specialized to solving feasible linear optimization problems with rational data, the ellipsoid method is an algorithm which finds an optimal solution in a finite number of steps.
The ellipsoid method generates a sequence of ellipsoids whose volume uniformly decreases at every step, thus enclosing a minimizer of a convex function.Figure of the Earth
The figure of the Earth is the size and shape of the Earth in geodesy. Its specific meaning depends on the way it is used and the precision with which the Earth's size and shape is to be defined. While the sphere is a close approximation of the true figure of the Earth and satisfactory for many purposes, geodesists have developed several models that more closely approximate the shape of the Earth so that coordinate systems can serve the precise needs of navigation, surveying, cadastre, land use, and various other concerns.Flattening
Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution (spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening is f and its definition in terms of the semi-axes of the resulting ellipse or ellipsoid is
The compression factor is b/a in each case. For the ellipse, this factor is also the aspect ratio of the ellipse.
There are two other variants of flattening (see below) and when it is necessary to avoid confusion the above flattening is called the first flattening. The following definitions may be found in standard texts and online web textsGeodesics 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 geodesic
is the shortest path between two points on a curved surface, i.e., the analogue
of a straight line on a plane surface. The solution of a triangulation
network on an ellipsoid is therefore a set of exercises in spheroidal
trigonometry (Euler 1755).
If the Earth is treated as a sphere, the geodesics are
great circles (all of which are closed) and the problems reduce to
ones in spherical trigonometry. However, Newton (1687)
showed that the effect of the rotation of the Earth results in its
resembling a slightly oblate ellipsoid and, in this case, the
equator and the meridians are the only simple
closed geodesics. Furthermore, the shortest path between two points on
the equator does not necessarily run along the equator. Finally, if the
ellipsoid is further perturbed to become a triaxial ellipsoid (with
three distinct semi-axes), only three geodesics are closed.Geodesy
Geodesy (), is the Earth science of accurately measuring and understanding Earth's geometric shape, orientation in space, and gravitational field. The field also incorporates studies of how these properties change over time and equivalent measurements for other planets (known as planetary geodesy). Geodynamical phenomena include crustal motion, tides, and polar motion, which can be studied by designing global and national control networks, applying space and terrestrial techniques, and relying on datums and coordinate systems.Geodetic datum
A geodetic datum or geodetic system (also: geodetic reference datum or geodetic reference system) is a coordinate system, and a set of reference points, used to locate places on the Earth (or similar objects). An approximate definition of sea level is the datum WGS 84, an ellipsoid, whereas a more accurate definition is Earth Gravitational Model 2008 (EGM2008), using at least 2,159 spherical harmonics. Other datums are defined for other areas or at other times; ED50 was defined in 1950 over Europe and differs from WGS 84 by a few hundred meters depending on where in Europe you look.
Mars has no oceans and so no sea level, but at least two martian datums have been used to locate places there.
Datums are used in geodesy, navigation, and surveying by cartographers and satellite navigation systems to translate positions indicated on maps (paper or digital) to their real position on Earth. Each starts with an ellipsoid (stretched sphere), and then defines latitude, longitude and altitude coordinates. One or more locations on the Earth's surface are chosen as anchor "base-points".
The difference in co-ordinates between datums is commonly referred to as datum shift. The datum shift between two particular datums can vary from one place to another within one country or region, and can be anything from zero to hundreds of meters (or several kilometers for some remote islands). The North Pole, South Pole and Equator will be in different positions on different datums, so True North will be slightly different. Different datums use different interpolations for the precise shape and size of the Earth (reference ellipsoids).
Because the Earth is an imperfect ellipsoid, localised datums can give a more accurate representation of the area of coverage than WGS 84. OSGB36, for example, is a better approximation to the geoid covering the British Isles than the global WGS 84 ellipsoid. However, as the benefits of a global system outweigh the greater accuracy, the global WGS 84 datum is becoming increasingly adopted.Horizontal datums are used for describing a point on the Earth's surface, in latitude and longitude or another coordinate system. Vertical datums measure elevations or depths.Geographic coordinate system
A geographic coordinate system is a coordinate system that enables every location on Earth to be specified by a set of numbers, letters or symbols. The coordinates are often chosen such that one of the numbers represents a vertical position and two or three of the numbers represent a horizontal position; alternatively, a geographic position may be expressed in a combined three-dimensional Cartesian vector.
A common choice of coordinates is latitude, longitude and elevation.
To specify a location on a plane requires a map projection.Geoid
The geoid () is the shape that the ocean surface would take under the influence of the gravity and rotation of Earth alone, if other influences such as winds and tides were absent. This surface is extended through the continents (such as with very narrow hypothetical canals). According to Gauss, who first described it, it is the "mathematical figure of the Earth", a smooth but irregular surface whose shape results from the uneven distribution of mass within and on the surface of Earth. It can be known only through extensive gravitational measurements and calculations. Despite being an important concept for almost 200 years in the history of geodesy and geophysics, it has been defined to high precision only since advances in satellite geodesy in the late 20th century.
All points on a geoid surface have the same effective potential (the sum of gravitational potential energy and centrifugal potential energy). The force of gravity acts everywhere perpendicular to the geoid, meaning that plumb lines point perpendicular and water levels parallel to the geoid if only gravity and rotational acceleration were at work. The surface of the geoid is higher than the reference ellipsoid wherever there is a positive gravity anomaly (mass excess) and lower than the reference ellipsoid wherever there is a negative gravity anomaly (mass deficit).Latitude
In geography, latitude is a geographic coordinate that specifies the north–south position of a point on the Earth's surface. Latitude is an angle (defined below) which ranges from 0° at the Equator to 90° (North or South) at the poles. Lines of constant latitude, or parallels, run east–west as circles parallel to the equator. Latitude is used together with longitude to specify the precise location of features on the surface of the Earth. On its own, the term latitude should be taken to be the geodetic latitude as defined below. Briefly, geodetic latitude at a point is the angle formed by the vector perpendicular (or normal) to the ellipsoidal surface from that point, and the equatorial plane. Also defined are six auxiliary latitudes which are used in special applications.Map projection
A map projection is a systematic transformation of the latitudes and longitudes of locations from the surface of a sphere or an ellipsoid into locations on a plane. Maps cannot be created without map projections. All map projections necessarily distort the surface in some fashion. Depending on the purpose of the map, some distortions are acceptable and others are not; therefore, different map projections exist in order to preserve some properties of the sphere-like body at the expense of other properties. There is no limit to the number of possible map projections.More generally, the surfaces of planetary bodies can be mapped even if they are too irregular to be modeled well with a sphere or ellipsoid; see below. Even more generally, projections are a subject of several pure mathematical fields, including differential geometry, projective geometry, and manifolds. However, "map projection" refers specifically to a cartographic projection.Map projection of the tri-axial ellipsoid
In geodesy, a map projection of the tri-axial ellipsoid maps Earth or some other astronomical body modeled as a tri-axial ellipsoid to the plane. Such a model is called the reference ellipsoid. In most cases, reference ellipsoids are spheroids, and sometimes spheres. Massive objects have sufficient gravity to overcome their own rigidity and usually have an oblate ellipsoid shape. However, minor moons or small solar system bodies are not under hydrostatic equilibrium. Usually such bodies have irregular shapes. Furthermore, some of gravitationally rounded objects may have a tri-axial ellipsoid shape due to rapid rotation (such as Haumea) or unidirectional strong tidal forces (such as Io).Ordnance Survey National Grid
The Ordnance Survey National Grid reference system is a system of geographic grid references used in Great Britain, distinct from latitude and longitude. It is often called British National Grid (BNG).The Ordnance Survey (OS) devised the national grid reference system, and it is heavily used in their survey data, and in maps based on those surveys, whether published by the Ordnance Survey or by commercial map producers. Grid references are also commonly quoted in other publications and data sources, such as guide books and government planning documents.
A number of different systems exist that can provide grid references for locations within the British Isles: this article describes the system created solely for Great Britain and its outlying islands (including the Isle of Man); the Irish grid reference system was a similar system created by the Ordnance Survey of Ireland and the Ordnance Survey of Northern Ireland for the island of Ireland. The Universal Transverse Mercator coordinate system (UTM) is used to provide grid references for worldwide locations, and this is the system commonly used for the Channel Islands and Ireland (since 2001). European-wide agencies also use UTM when mapping locations, or may use the Military Grid Reference System (MGRS) system, or variants of it. OSGB uses Orthorectified images of many temporal resolution for one area.Reference ellipsoid
In geodesy, a reference ellipsoid is a mathematically defined surface that approximates the geoid, the truer figure of the Earth, or other planetary body.
Because of their relative simplicity, reference ellipsoids are used as a preferred surface on which geodetic network computations are performed and point coordinates such as latitude, longitude, and elevation are defined.
In the context of standardization and geographic applications, a geodesic reference ellipsoid is the mathematical model used as foundation by Spatial reference system or Geodetic datum definitions.SK-42 reference system
The SK-42 reference system also known as the Krasovsky 1940 ellipsoid, is a coordinate system established in the Soviet Union in 1942 as Systema koordinat (Russian: Система координат 1942 года), and provides parameters which are linked to the geocentric Cartesian coordinate system PZ-90 . It was used in geodetic calculations, notably in military mapping and determining state borders.
The coordinate system SK-42 served as a foundation for developing the SK-63 reference system which was created and used primarily for civilian and industrial development purposes..
The Krasovsky 1940 ellipsoid uses a semi-major axis (equatorial radius) a of 6,378,245 m, and an inverse flattening 1/f of 298.3Spheroid
A spheroid, or ellipsoid of revolution, 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 circular symmetry.
If the ellipse is rotated about its major axis, the result is a prolate (elongated) spheroid, shaped like an American football or rugby ball. If the ellipse is rotated about its minor axis, the result is an oblate (flattened) spheroid, shaped like a lentil. If the generating ellipse is a circle, the result is a sphere.
Due to the combined effects of gravity and rotation, the figure of the Earth (and of all planets) is not quite a sphere, but instead is slightly flattened in the direction of its axis of rotation. For that reason, in cartography the Earth is often approximated by an oblate spheroid instead of a sphere. The current World Geodetic System model uses a spheroid whose radius is 6,378.137 km (3,963.191 mi) at the Equator and 6,356.752 km (3,949.903 mi) at the poles.
The word spheroid originally meant "an approximately spherical body", admitting irregularities even beyond the bi- or tri-axial ellipsoidal shape, and that is how the term is used in some older papers on geodesy (for example, referring to truncated spherical harmonic expansions of the Earth).World Geodetic System
The World Geodetic System (WGS) is a standard for use in cartography, geodesy, and satellite navigation including GPS. It comprises a standard coordinate system for the Earth, a standard spheroidal reference surface (the datum or reference ellipsoid) for raw altitude data, and a gravitational equipotential surface (the geoid) that defines the nominal sea level.
The latest revision is WGS 84 (also known as WGS 1984, EPSG:4326), established in 1984 and last revised in 2004. Earlier schemes included WGS 72, WGS 66, and WGS 60. WGS 84 is the reference coordinate system used by the Global Positioning System.