Ellipsoid

Ellipsoid

Overview


An ellipsoid is a closed type of quadric surface
Quadric
In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...

 that is a higher dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

al analogue of an ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

. The equation of a standard axis-aligned ellipsoid body in an xyz-Cartesian coordinate system
Cartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

 is


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 number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s determining the shape of the ellipsoid.

More generally, an arbitrarily oriented ellipsoid, centered at v, is defined by the equation


where A is a positive definite matrix and x, v are vectors.
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Encyclopedia


An ellipsoid is a closed type of quadric surface
Quadric
In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...

 that is a higher dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

al analogue of an ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

. The equation of a standard axis-aligned ellipsoid body in an xyz-Cartesian coordinate system
Cartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

 is


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 number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s determining the shape of the ellipsoid.

More generally, an arbitrarily oriented ellipsoid, centered at v, is defined by the equation


where A is a positive definite matrix and x, v are vectors. In that case, the eigenvectors of A define the principal directions of the ellipsoid and the square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

 of the eigenvalues are the corresponding equatorial radii.

If all three radii are equal, the solid body is a sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

; if two radii are equal, the ellipsoid is a spheroid
Spheroid
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....

:
  • Sphere
    Sphere
    A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

    ;
  • Oblate spheroid (disk-shaped);
  • Prolate
    Prolate spheroid
    A prolate spheroid is a spheroid in which the polar axis is greater than the equatorial diameter. Prolate spheroids stand in contrast to oblate spheroids...

    spheroid (like a rugby ball);
  • 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
Semi-major axis
The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...

 and semi-minor axis
Semi-minor axis
In geometry, the semi-minor axis is a line segment associated with most conic sections . One end of the segment is the center of the conic section, and it is at right angles with the semi-major axis...

 of the appropriate ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

s.

Scalene ellipsoids are frequently called "triaxial ellipsoids", the implication being that all three axes need to be specified to define the shape.

Any planar cross section
Cross section (geometry)
In geometry, a cross-section is the intersection of a figure in 2-dimensional space with a line, or of a body in 3-dimensional space with a plane, etc...

 passing through the center of an ellipsoid forms an ellipse on its surface, with the possible special case of a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

 if the three radii are the same (i.e., the ellipsoid is a sphere) or if the plane is parallel to two radii that are equal.

Parameterization


Using the common coordinates, where is a point's reduced, or parametric latitude and is its planetographic longitude
Longitude
Longitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....

, an ellipsoid can be parameterized by:
(Note that this parameterization is not one-to-one at the poles, where )

Volume


The volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

 of an ellipsoid is given by the formula
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
Prolate spheroid
A prolate spheroid is a spheroid in which the polar axis is greater than the equatorial diameter. Prolate spheroids stand in contrast to oblate spheroids...

 spheroid
Spheroid
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....

 when two of them are equal.
The volumes of the maximum inscribed and minimum circumscribed boxes are respectively: and .

Surface area


The surface area
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

 of an ellipsoid is given by:
where
is the modular angle, or angular eccentricity
Angular eccentricity
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....

; and , are the incomplete elliptic integral
Elliptic integral
In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler...

s of the first and second kind.

Unlike the surface area of a sphere, the surface area of a general ellipsoid cannot be expressed exactly by an elementary function.

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 circular equator):
 If oblate:
If prolate:


In the "flat" limit of , the area is approximately

Mass properties


The mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

 of an ellipsoid of uniform density is:
where is the density.

The mass moments of inertia
Moment of inertia
In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...

 of an ellipsoid of uniform density are:


where , , and are the moments of inertia about the x, y, and z axes, respectively. Products of inertia
Moment of inertia
In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...

 are zero.

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 principal inertias of an arbitrary rigid body are known, an equivalent ellipsoid of uniform density can be constructed, with the following characteristics:

Rotational equilibrium



Scalene ellipsoids and cuboid
Cuboid
In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing definitions of a cuboid in mathematical literature...

s 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
Moment of inertia
In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...

 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 generally rotate along their minor axes (as does the Earth, which is merely oblate); in addition, because of tidal locking
Tidal locking
Tidal locking occurs when the gravitational gradient makes one side of an astronomical body always face another; for example, the same side of the Earth's Moon always faces the Earth. A tidally locked body takes just as long to rotate around its own axis as it does to revolve around its partner...

, scalene moons in synchronous orbit
Synchronous orbit
A synchronous orbit is an orbit in which an orbiting body has a period equal to the average rotational period of the body being orbited , and in the same direction of rotation as that body.-Properties:...

 such as Mimas
Mimas (moon)
Mimas is a moon of Saturn which was discovered in 1789 by William Herschel. It is named after Mimas, a son of Gaia in Greek mythology, and is also designated Saturn I....

 orbit with their major axis aligned radially to their planet.

A relaxed ellipsoid, that is, one in hydrostatic equilibrium
Hydrostatic equilibrium
Hydrostatic equilibrium or hydrostatic balance is the condition in fluid mechanics where a volume of a fluid is at rest or at constant velocity. This occurs when compression due to gravity is balanced by a pressure gradient force...

, has an oblateness directly proportional to its mean density and mean radius. Ellipsoids with a differentiated interior—that is, a denser core than mantle—have a lower oblateness than a homogeneous body. Over all, the ratio (b–c)/(a−c) is approximately 0.25, though this drops for rapidly rotating bodies.

Fluid properties


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.

Linear transformations


An invertible linear transformation
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...

 applied to a sphere produces an ellipsoid, which can be brought into the above standard form by a suitable rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

, a consequence of the polar decomposition (also, see spectral theorem
Spectral theorem
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 broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...

). 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. The singular value decomposition
Singular value decomposition
In linear algebra, the singular value decomposition is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics....

 and polar decomposition are matrix decompositions closely related to these geometric observations.

The intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....

 of an ellipsoid with a plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

 is either empty
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...

, a single point, or an ellipse (including a circle).

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
Egg (food)
Eggs are laid by females of many different species, including birds, reptiles, amphibians, and fish, and have probably been eaten by mankind for millennia. Bird and reptile eggs consist of a protective eggshell, albumen , and vitellus , contained within various thin membranes...

 is approximately that of half each a prolate and a roughly spherical (potentially even minorly oblate) ellipsoid joined at the equator, sharing a principal axis of rotational symmetry
Rotational symmetry
Generally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted, the triskelion appearing on the Isle of Man's flag has...

. Although the term egg-shaped usually implies a lack of reflection symmetry
Reflection symmetry
Reflection symmetry, reflectional symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.In 2D there is a line of symmetry, in 3D a...

 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 one of the 3D surfaces described above. See also oval
Oval
An oval is any curve resembling an egg or an ellipse, such as a Cassini oval. The term does not have a precise mathematical definition except in one area oval , but it may also refer to:* A sporting arena of oval shape** a cricket field...

.

See also

  • Paraboloid
    Paraboloid
    In mathematics, a paraboloid is a quadric surface of special kind. There are two kinds of paraboloids: elliptic and hyperbolic. The elliptic paraboloid is shaped like an oval cup and can have a maximum or minimum point....

  • Hyperboloid
  • Reference ellipsoid
    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....

  • Geoid
    Geoid
    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 continents . According to C.F...

  • Ellipsoid method
    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...

  • Superellipsoid
    Superellipse
    A superellipse is a geometric figure defined in the Cartesian coordinate system as the set of all points withwhere n, a and b are positive numbers....

    , an ellipsoid-shaped planetoid
  • Homoeoid
    Homoeoid
    A homoeoid is a shell bounded by two concentric, similar ellipses or ellipsoids.When the thickness of the shell becomes negligible, it is called a thin homoeoid.-Mathematical definition:If the outer shell is given by...

    , a shell bounded by two concentric, similar ellipsoids
  • Focaloid, a shell bounded by two concentric, confocal ellipsoids
  • Elliptical distribution
    Elliptical distribution
    In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution and inherit some of its properties.-Definition:...

    , in statistics
  • Ellipse
    Ellipse
    In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...


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