Ellipsoid

This article is about the shape. For the type of theatrical spotlight, see ellipsoidal reflector spotlight. Ellipsoidal reflector spotlight (abbreviated to ERS, or colloquially ellipsoidal) is the name for a type of Stage lighting instrument, named for the

Shaded wireframe rendering of an ellipsoid with a = 3, b = 2, c = 1 (scalene ellipsoid).

Wireframe rendering of an ellipsoid (oblate spheroid)

An ellipsoid is a type of quadric surface that is a higher dimensional analogue of an ellipse. In mathematics a quadric, or quadric surface, is any D -dimensional Hypersurface defined as the locus of zeros of a Quadratic In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a The equation of a standard ellipsoid body in an xyz-Cartesian coordinate system is

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

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. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane A negative number is a Number that is less than zero, such as −2 In Mathematics, the real numbers may be described informally in several different ways

If all three radii are equal, the solid body is a sphere; if two radii are equal, the ellipsoid is a spheroid:

$a=b=c:\,\!$ Sphere;
$a=b>c:\,\!$ Oblate spheroid (disk-shaped);
$a=b<c:\,\!$ Prolate spheroid (cigar-shaped);
$a>b>c:\,\!$ Scalene ellipsoid ("three unequal sides"). "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe Equation A spheroid centered at the origin and rotated about the z axis is defined by the implicit equation \left(\frac{x}{a}\right^2+\left(\frac{y}{a}\right^2+\left(\frac{z}{b}\right^2 "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe An oblate Spheroid is a rotationally symmetric Ellipsoid having a polar axis shorter than the diameter of the equatorial circle whose plane A prolate spheroid is a Spheroid in which the polar Diameter is longer than the Equatorial diameter

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. In Geometry, the 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) is a Line segment associated with most Conic sections (that is with ellipses and In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a

1 Parameterization
2 Volume
3 Surface area
4 Mass properties
5 Linear transformations
6 Egg shape
7 References
8 See also
9 External links

Parameterization

Using the common coordinates, where $\beta\,\!$ is a point's reduced, or parametric latitude and ${\color{white}+}\!\!\!\lambda{\color{white}'}\,\!$ is its planetographic longitude, an ellipsoid can be parameterized by:

$\begin{align} x&=a\,\cos(\beta)\cos(\lambda);\!{\color{white}|}\\ y&=b\,\cos(\beta)\sin(\lambda);\\ z&=c\,\sin(\beta);\end{align}\,\!$

$\begin{matrix}-90^\circ\leq\beta\leq+90^\circ; \quad-180^\circ\leq\lambda\leq+180^\circ;\!{\color{white}\big|}\end{matrix}\,\!$

(Note that this parameterization is not

1-1 at the poles, where $\scriptstyle{{\color{white}|}\beta=\pm{90}^\circ}{\color{white}|}\,\!$ )

Or, using spherical coordinates, where ${\color{white}+}\!\!\!\theta{\color{white}'}\,\!$ is the colatitude, or zenith, and ${\color{white}+}\!\!\!\varphi{\color{white}\!\!\!-}\,\!$ is the longitude in 360°, or azimuth:

$\begin{align} x&=a\,\sin(\theta)\cos(\varphi);\!{\color{white}|}\\ y&=b\,\sin(\theta)\sin(\varphi);\\ z&=c\,\cos(\theta);\end{align}\,\!$

$\begin{matrix}0\leq\theta\leq{180}^\circ; \quad{0}\leq\varphi\leq{360}^\circ;\!{\color{white}\big|}\end{matrix}\,\!$

Volume

The volume of an ellipsoid is given by the formula

$\frac{4}{3}\pi abc.\,\!$

Note that this equation reduces to that of the volume of a sphere when all three elliptic radii are equal, and to that of an oblate or prolate spheroid when two of them are equal. Latitude, usually denoted symbolically by the Greek letter phi ( Φ) gives the location of a place on Earth (or other planetary body north or south of the Longitude (ˈlɒndʒɪˌtjuːd or ˈlɒŋgɪˌtjuːd symbolized by the Greek character Lambda (λ is the east-west Geographic coordinate measurement In Mathematics, the spherical coordinate system is a Coordinate system for representing geometric figures in three dimensions using three coordinates the radial The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically An oblate Spheroid is a rotationally symmetric Ellipsoid having a polar axis shorter than the diameter of the equatorial circle whose plane A prolate spheroid is a Spheroid in which the polar Diameter is longer than the Equatorial diameter Equation A spheroid centered at the origin and rotated about the z axis is defined by the implicit equation \left(\frac{x}{a}\right^2+\left(\frac{y}{a}\right^2+\left(\frac{z}{b}\right^2

Surface area

The surface area of an ellipsoid is given by:

$2\pi\left(c^2+b\sqrt{a^2-c^2}E(o\!\varepsilon,m)+\frac{bc^2}{\sqrt{a^2-c^2}}F(o\!\varepsilon,m)\right),\,\!$

where

$o\!\varepsilon=\arccos\left(\frac{c}{a}\right)\;\textrm{(oblate)\;\;or\;\;}\arccos\left(\frac{a}{c}\right)\;(\textrm{prolate}),\,\!$

is the modular angle, or angular eccentricity; $m=\frac{b^2-c^2}{b^2\sin(o\!\varepsilon)^2}\,\!$ and $E(o\!\varepsilon,m)\,\!$ , $F(o\!\varepsilon,m)\,\!$ are the incomplete elliptic integrals of the first and second kind. Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. 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.

Unlike the area of a sphere, the surface area of a general ellipsoid cannot be expressed exactly by an elementary function. This article discusses the concept of elementary functions in differential algebra

An approximate formula is:

$\approx 4\pi\!\left(\frac{ a^p b^p+a^p c^p+b^p c^p }{3}\right)^{1/p}.\,\!$

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).

Exact formulae can be obtained for the case a = b (i. e. , a spherical equator):

If oblate: $2\pi\!\left(a^2+c^2\frac{\operatorname{arctanh}(\sin(o\!\varepsilon))}{\sin(o\!\varepsilon)}\right);\,\!$

If prolate: $2\pi\!\left(a^2+c^2\frac{o\!\varepsilon}{\tan(o\!\varepsilon)}\right);\,\!$

In the "flat" limit of $c \ll a, b\,\!$ , the area is approximately $2\pi ab.\,\!$

Mass properties

The mass of an ellipsoid of uniform density is:

$m = \rho V = \rho \frac{4}{3} \pi abc\,\!$

where $\rho\,\!$ is the density. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object

The mass moments of inertia of an ellipsoid of uniform density are:

$I_{\mathrm{xx}} = m {b^2+c^2 \over 5}$

$I_{\mathrm{yy}} = m {c^2+a^2 \over 5}$

$I_{\mathrm{zz}} = m {a^2+b^2 \over 5}$

where $I_{\mathrm{xx}}\,\!$ , $I_{\mathrm{yy}}\,\!$ , and $I_{\mathrm{zz}}\,\!$ are the moments of inertia about the x, y, and z axes, respectively. This article is about the moment of inertia of a rotating object. Products of inertia are zero. This article is about the moment of inertia of a rotating object.

It can easily be shown that if a=b=c, then the moments of inertia reduce to those for a uniform-density sphere.

Conversely, if the mass and principle inertias of an arbitrary rigid body are known, an equivalent ellipsoid of uniform density can be constructed, with the following characteristics:

$a = \sqrt{{5 \over 2} {I_{\mathrm{yy}}+I_{\mathrm{zz}}-I_{\mathrm{xx}} \over m}}$

$b = \sqrt{{5 \over 2} {I_{\mathrm{zz}}+I_{\mathrm{xx}}-I_{\mathrm{yy}} \over m}}$

$c = \sqrt{{5 \over 2} {I_{\mathrm{xx}}+I_{\mathrm{yy}}-I_{\mathrm{zz}} \over m}}$

$\rho = \frac{3}{4} {m \over \pi abc}\!$

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. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation In Mathematics, particularly Linear algebra and Functional analysis, the spectral theorem is any of a number of results about Linear operators 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 Linear algebra, a symmetric matrix is a Square matrix, A, that is equal to its Transpose A = A^{T}

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 In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members

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

Oval

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. An egg is a round or oval body laid by the female of many animals consisting of an Ovum surrounded by layers of Membranes and an outer casing which acts to nourish Generally speaking an object with rotational symmetry is an object that looks the same after a certain amount of Rotation. ^[1] Although the term egg-shaped usually implies a lack of reflection symmetry across the equatorial plane, it may also refer to true prolate ellipsoids. Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is Symmetry with respect It can also be used to describe the 2D figure that, revolved around its major axis, produces the 3D surface. In Geometry, the semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae See also oval (geometry). In geometry an oval or ovoid (from Latin ovum, 'egg' is any Curve resembling an egg or an Ellipse.

References

^ Egg Curves by Jürgen Köller.

"Ellipsoid" by Jeff Bryant, The Wolfram Demonstrations Project, 2007.
Ellipsoid and Quadratic Surface, MathWorld. MathWorld is an online Mathematics reference work created and largely written by Eric W

External links

Interactive Java 3D model of the ellipsoid

In Mathematics, a paraboloid is a Quadric surface of special kind In Mathematics, a hyperboloid is a Quadric, a type of surface in three Dimensions described by the equation {x^2 \over a^2} + 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 geoid is that Equipotential surface which would coincide exactly with the mean ocean surface of the Earth if the oceans were in equilibrium at rest and extended through The ellipsoid method is an Algorithm for solving Convex optimization problems The superellipse (or Lamé curve) is the geometric figure defined in the Cartesian coordinate system as the set of all points ( x, y) with

Dictionary

-noun

(mathematics) a surface, all of whose cross sections are elliptic or circular (includes the sphere)
(geography) Such a surface used as a model of the shape of the earth.

-adjective

(mathematics) of or pertaining to an ellipse; ellipsoidal
Shaped like an ellipse; elliptical.
Shaped like a symmetrical oval that is evenly tapered on both ends.

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