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Encyclopedia > Ellipsoid

3D rendering of an ellipsoid
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3D rendering of an ellipsoid

In mathematics, an ellipsoid is a type of quadric that is a higher dimensional analogue of an ellipse. The equation of a standard ellipsoid in an x-y-z Cartesian coordinate system is ImageMetadata File history File links Ellipsoid_3d. ... ImageMetadata File history File links Ellipsoid_3d. ... Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... 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. ... The ellipse and some of its mathematical properties. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

{x^2 over a^2}+{y^2 over b^2}+{z^2 over c^2}=1

where a, b and c (the lengths of the three semi-axes) are fixed positive real numbers determining the shape of the ellipsoid. If two of those numbers are equal, the ellipsoid is a spheroid; if all three are equal, it is a sphere. A negative number is a number that is less than zero, such as −3. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, a spheroid is a quadric surface in three dimensions obtained by rotating an ellipse about one of its principal axes. ... A sphere (< Greek σφαίρα) is a perfectly symmetrical geometrical object. ...


If we assume a ≥ b ≥ c, then when:

  • a ≠ b ≠ c we have a scalene ellipsoid
  • In the limit c tending to 0 it is an ellipse
  • a > b = c the ellipsoid is a prolate spheroid (cigar-shaped)
  • a = b > c the ellipsoid is an oblate spheroid (disk-shaped)
  • a = b = c we have a sphere, as aforementioned

Contents

A spheroid is a quadric surface in three dimensions obtained by rotating an ellipse about one of its principal axes. ... An oblate spheroid is ellipsoid having a shorter axis and two equal longer axes. ...


Parametrization

An ellipsoid can be parametrized by:

x = a,sin phi cos theta
y = b,sin phi sin theta
z = c,cos phi
0 leq theta < 2pi
0 leq phi leq pi

Note that this parametrization is not 1-1 at the points where φ = 0,π.


Volume

The volume of an ellipsoid is given by: GEE GUY dimensions is called content. ...

frac{4}{3} pi abc

Surface area

The surface area of an ellipsoid is given by: Area is a physical quantity expressing the size of a part of a surface. ...

2 pi left( c^2 + frac{bc^2}{sqrt{a^2-c^2}} F(theta, m) + bsqrt{a^2-c^2} E(theta, m) right)

where

m = frac{a^2(b^2-c^2)}{b^2(a^2-c^2)}
theta = arcsin{left( e right)}
e = sqrt{1 - frac{c^2}{a^2}}

and F(θ,m) and E(θ,m) are the incomplete elliptic integrals of the first and second kind. 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. ...


Exact formulae are:

If flat: = 2 pi left( ab right)
If prolate: = 2 pi left( c^2 + ac frac{arcsin{left( e right)}}{e} right)
If oblate: = 2 pi left( a^2 + c^2 frac{operatorname{arctanh}{left( e right)}}{e} right)

Approximate formula is:

If scalene: approx 4 pi left( frac{ a^p b^p + a^p c^p + b^p c^p }{3} right)^{1/p}

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).


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. ... Rotation of a plane, seen as the rotation of the terrain relative to the plane (exposure time 1. ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection 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. ... Two intersecting planes in R3 In mathematics, a plane is a fundamental two-dimensional object. ... In mathematics and more specifically set theory, the empty set is the unique set which contains 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 an egg is approximately an oblate ellipsoid, but, while keeping cylindrical symmetry, there is not quite symmetry in a plane perpendicular to the long axis. The term egg-shaped is typically used taking this asymmetry into account, but it may also simply mean oblate ellipsoid. It can also be used for a 2D shape. See also oval (geometry). Image File history File links Oval1. ... Image File history File links Oval1. ... A carton of free-range chicken eggs Ostrich egg Bird eggs are a common food source. ... In geometry, an oval or ovoid (from Latin ovum, egg) is any curve resembling an egg or an ellipse. ...


See also


  Results from FactBites:
 
Index ellipsoid - Wikipedia, the free encyclopedia (188 words)
The index ellipsoid is a diagram of an ellipsoid that depicts the orientation and relative magnitude of refractive indices in a crystal.
The equation for the ellipsoid is constructed using the electric displacement vector, D, and the dielectric constants.
The semiaxes of this ellipsoid are dielectric constants of the crystal.
ellipsoid - definition of ellipsoid in Encyclopedia (318 words)
In mathematics, an ellipsoid is a type of quadric that is a higher dimensional analogue of an ellipse.
If two of those numbers are equal, the ellipsoid is a spheroid; if all three are equal, we have a sphere.
The intersection of an ellipsoid with a plane is empty, a single point or an ellipse.
  More results at FactBites »

 

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