Examples of ellipsoids with equation x2/a2 + y2/b2 + z2/c2 = 1:
Sphere, a = b = c = 4, top;
Spheroid, a = b = 5, c = 3, bottom left;
Tri-axial ellipsoid, a = 4.5, b = 6; c = 3, bottom right
An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
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 figure is a triaxial ellipsoid (rarely scalene ellipsoid), 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.
An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation...
An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy...
the angle formed between the vector perpendicular (or normal) to the ellipsoidal surface from the point, and the plane of the equator. Two levels of abstraction...
the ellipsoid method is an iterative method for minimizing convex functions over convex sets. The ellipsoid method generates a sequence of ellipsoids whose...
A Jacobi ellipsoid is a triaxial (i.e. scalene) ellipsoid under hydrostatic equilibrium which arises when a self-gravitating, fluid body of uniform density...
The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth...
In geodesy, the Hayford ellipsoid is a reference ellipsoid named after the American geodesist John Fillmore Hayford (1868–1925), which was introduced in...
Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system ( λ , μ , ν ) {\displaystyle (\lambda ,\mu ,\nu )} that generalizes the two-dimensional...
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...
ellipse. There are two types of ellipsoidal domes: prolate ellipsoidal domes and oblate ellipsoidal domes. A prolate ellipsoidal dome is derived by rotating...
Erdmessung (Central Bureau of International Geodesy), and a series of global ellipsoids of the Earth were derived (e.g., Helmert 1906, Hayford 1910/ 1924). A...
John ellipsoid or Löwner–John ellipsoid E(K) associated to a convex body K in n-dimensional Euclidean space Rn can refer to the n-dimensional ellipsoid of...
celestial body is called a reference ellipsoid. The reference ellipsoid for Earth is called an Earth ellipsoid. An ellipsoid of revolution is uniquely defined...
geodesy based on a reference ellipsoid. They include geodetic latitude (north/south) ϕ, longitude (east/west) λ, and ellipsoidal height h (also known as geodetic...
standards, also includes a choice of geodetic datum (including an Earth ellipsoid), as different datums will yield different latitude and longitude values...
The Bessel ellipsoid (or Bessel 1841) is an important reference ellipsoid of geodesy. It is currently used by several countries for their national geodetic...
Ordnance Survey maps. The Airy ellipsoid is a regional best fit for Britain; more modern mapping tends to use the GRS80 ellipsoid used by the Global Positioning...
reference ellipsoid is called geoidal undulation, and it varies globally between ±110 m based on the GRS 80 ellipsoid. A reference ellipsoid, customarily...
radius, which is the radius of a sphere having the same volume as the ellipsoid (R3). All three values are about 6,371 kilometres (3,959 mi). Other ways...
giving us a varying ellipsoid of constant energy. This is shown in the animation as a fixed orange ellipsoid and increasing blue ellipsoid. For concreteness...
A condyloid joint (also called condylar, ellipsoidal, or bicondylar) is an ovoid articular surface, or condyle that is received into an elliptical cavity...
precise reference ellipsoids. For rigid-surface nearly-spherical bodies, which includes all the rocky planets and many moons, ellipsoids are defined in terms...
In crystal optics, the index ellipsoid (also known as the optical indicatrix or sometimes as the dielectric ellipsoid) is a geometric construction which...
level (MSL). Since the rise of the global positioning system (GPS), the ellipsoid and datum WGS 84 it uses has supplanted most others in many applications...
The manipulability ellipsoid is a concept in robotics that represents the manipulability of a robotic system in a graphical form. Here, the manipulability...
Thermal ellipsoids, more formally termed atomic displacement parameters or anisotropic displacement parameters, are ellipsoids used in crystallography...
In geometry, ellipsoid packing is the problem of arranging identical ellipsoid throughout three-dimensional space to fill the maximum possible fraction...
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