3D rendering of an ellipsoid (prolate spheroid)
Wireframe rendering of an ellipsoid (oblate spheroid) An ellipsoid is a type of quadric surface that is a higher dimensional analogue of an ellipse. The equation of a standard ellipsoid body in an x-y-z Cartesian coordinate system is A Lekolite ERS. The name Leko can refer to any ERS An Elipsoidal Reflector from a Leko Source Four ERS Ellipsoidal reflector spotlight (abbreviated to ERS, or colloquially ellipsoidal) is the name for a type of theatrical light, getting the name from the ellipsoidal reflector used to intensify the light...
ImageMetadata File history File links Ellipsoid_3d. ...
ImageMetadata File history File links Ellipsoid_3d. ...
Image File history File links Gnuplot_ellipsoid. ...
Image File history File links Gnuplot_ellipsoid. ...
Ellipsoid Elliptic Paraboloid Hyperbolic Paraboloid Hyperboloid of One Sheet Hyperboloid of Two Sheets Cone Elliptic Cylinder Hyperbolic Cylinder Parabolic Cylinder In mathematics a quadric, or quadric surface, is any D-dimensional (hyper-)surface represented by a second-order equation in spatial variables (coordinates). ...
2-dimensional renderings (ie. ...
For other uses, see Ellipse (disambiguation). ...
Fig. ...
where a and b are the equatorial radii (along the x and y axes) and c is the polar radius (along the z-axis), all of which are fixed positive real numbers determining the shape of the ellipsoid. A negative number is a number that is less than zero, such as −3. ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
If all three radii are equal, the solid body is a sphere; if two radii are equal, the ellipsoid is a spheroid: A sphere is a symmetrical geometrical object. ...
In mathematics, a spheroid is a quadric surface in three dimensions obtained by rotating an ellipse about one of its principal axes. ...
- Sphere;
- Oblate spheroid (disk-shaped);
- Prolate spheroid (cigar-shaped);
- Scalene ellipsoid ("three unequal sides").
The points (a,0,0), (0,b,0) and (0,0,c) lie on the surface and the line segments from the origin to these points are called the semi-principal axes. These correspond to the semi-major axis and semi-minor axis of the appropriate ellipses. A sphere is a symmetrical geometrical object. ...
An oblate spheroid is ellipsoid having a shorter axis and two equal longer axes. ...
The semi-major axis of an ellipse In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae. ...
In geometry, the semi-minor axis (also semiminor axis) applies to ellipses and hyperbolas. ...
For other uses, see Ellipse (disambiguation). ...
Parameterization
Where is a point's parametric latitude and is its planetographic longitude, an ellipsoid can be parameterized by: -
(Note that this parameterization is not 1-1 at the points where .)
Volume The volume of an ellipsoid is given by (note that this equation reduces to that of the volume of a sphere when all three elliptic radii are equal): The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. ...
Surface area The surface area of an ellipsoid is given by: This article is about the physical quantity. ...
where is the modular angle, or angular eccentricity; and , are the incomplete elliptic integrals of the first and second kind. In the study of ellipses and related geometry, various parameters in the distortion of a circle into an ellipse are identified and employed: Aspect ratio, flattening and eccentricity. ...
In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler. ...
Unlike the area of a sphere, the surface area of a general ellipsoid cannot be expressed exactly by an elementary function. In mathematics, several functions are important enough to deserve their own name. ...
An approximate formula is:
-
Where p ≈ 1.6075 yields a relative error of at most 1.061% (Knud Thomsen's formula); a value of p = 8/5 = 1.6 is optimal for nearly spherical ellipsoids, with a relative error of at most 1.178% (David W. Cantrell's formula).
Exact formulae can be obtained for the case a = b (i.e., a spherical equator):
- If oblate:
- If prolate:
In the "flat" limit of , the area is approximately
Mass properties The mass of an ellipsoid of uniform density is: This article or section is in need of attention from an expert on the subject. ...
where is the density.
The mass moments of inertia of an ellipsoid of uniform density are: Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg m², Former British units slug ft2), is the rotational analog of mass. ...
where , , and are the moments of inertia about the x, y, and z axes, respectively. Products of inertia are zero. Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg m², Former British units slug ft2), is the rotational analog of mass. ...
It can easily be shown that if a=b=c, then the moments of inertia reduce to those for a uniform sphere.
Linear transformations If one applies an invertible linear transformation to a sphere, one obtains an ellipsoid; it can be brought into the above standard form by a suitable rotation, a consequence of the 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 semiaxes are given by the eigenvalues. In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...
A sphere rotating around its axis. ...
In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. ...
In linear algebra, a symmetric matrix is a matrix that is its own transpose. ...
The intersection of an ellipsoid with a plane is empty, a single point or an ellipse. In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ...
This article is about the mathematical construct. ...
The empty set is the set containing no elements. ...
One can also define ellipsoids in higher dimensions, 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.
Egg shape The shape of a chicken egg is approximately that of half each a prolate and roughly spherical (potentially even minorly oblate) ellipsoid joined at the equator, sharing a principal axis of rotational symmetry. Although the term egg-shaped usually implies a lack of reflection symmetry across the equatorial plane, it may also refer to true prolate ellipsoids. It can also be used to describe the 2D figure that, revolved around its major axis, produces the 3D surface. See also oval (geometry). An egg is a body consisting of an ovum surrounded by layers of membranes and an outer casing of some type, which acts to nourish and protect a developing embryo. ...
In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ...
The triskelion appearing on the Isle of Man flag. ...
Figures with the axes of symmetry drawn in. ...
In geometry, the semi-major axis (also semimajor axis) a applies to ellipses and hyperbolas. ...
In geometry, an oval or ovoid (from Latin ovum, egg) is any curve resembling an egg or an ellipse. ...
See also In mathematics, a spheroid is a quadric surface in three dimensions obtained by rotating an ellipse about one of its principal axes. ...
Paraboloid of revolution Hyperbolic paraboloid In mathematics, a paraboloid is a quadric, a type of surface in three dimensions, described by the equation: (elliptic paraboloid), or (hyperbolic paraboloid). ...
Hyperboloid of one sheet Hyperboloid of two sheets In mathematics, a hyperboloid is a quadric, a type of surface in three dimensions, described by the equation (hyperboloid of one sheet), or (hyperboloid of two sheets) If, and only if, a = b, it is a hyperboloid of revolution. ...
In geodesy, a reference ellipsoid is a mathematically-defined surface that approximates the geoid, the truer figure of the Earth, or other planetary body. ...
The GOCE project will measure high-accuracy gravity gradients and provide an accurate geoid model based on the Earths gravity field. ...
The ellipsoid method is an algorithm for solving linear programs. ...
Squircle, the superellipse for n = 4, a = b = 1, approximates a chamfered square. ...
(also written (136108) 2003 EL61), nicknamed Santa, is a large Kuiper belt object, roughly one-third the mass of Pluto, discovered by Mike Browns group at Caltech in the United States and J. L. Ortiz et al. ...
External Links - Interactive Java 3D model of the ellipsoid
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