![Ellipsoide](https://upload.wikimedia.org/wikipedia/commons/3/33/Ellipsoide.svg)
An ellipsoid is a surface that may be obtained from a
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
by deforming it by means of directional
scaling
Scaling may refer to:
Science and technology
Mathematics and physics
* Scaling (geometry), a linear transformation that enlarges or diminishes objects
* Scale invariance, a feature of objects or laws that do not change if scales of length, energ ...
s, or more generally, of an
affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generally, ...
.
An ellipsoid is a
quadric surface
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...
; that is, a
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...
that may be defined as the
zero set
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or equi ...
of a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
of degree two in three variables. Among quadric surfaces, an ellipsoid is characterized by either of the two following properties. Every planar
cross section
Cross section may refer to:
* Cross section (geometry)
** Cross-sectional views in architecture & engineering 3D
*Cross section (geology)
* Cross section (electronics)
* Radar cross section, measure of detectability
* Cross section (physics)
**Abs ...
is either an
ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
, or is empty, or is reduced to a single point (this explains the name, meaning "ellipse-like"). It is
bounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
, which means that it may be enclosed in a sufficiently large sphere.
An ellipsoid has three pairwise
perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
axes of symmetry
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.
In 2D ther ...
which intersect at a
center of symmetry A fixed point of an isometry group is a point that is a fixed point for every isometry in the group. For any isometry group in Euclidean space the set of fixed points is either empty or an affine space.
For an object, any unique centre and, more ...
, called the center of the ellipsoid. The
line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
s 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
In political science, a revolution (Latin: ''revolutio'', "a turn around") is a fundamental and relatively sudden change in political power and political organization which occurs when the population revolts against the government, typically due ...
'', also called a ''
spheroid
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 axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has cir ...
''. In this case, the ellipsoid is invariant under a
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
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
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 axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circ ...
''; if it is longer, it is a ''
prolate spheroid
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 axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circu ...
''. If the three axes have the same length, the ellipsoid is a sphere.
Standard equation
The general ellipsoid, also known as triaxial ellipsoid, is a quadratic surface which is defined in
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
as:
:
where
,
and
are the length of the semi-axes.
The points
,
and
lie on the surface. The line segments from the origin to these points are called the principal semi-axes of the ellipsoid, because are half the length of the principal axes. They correspond to the
semi-major axis
In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the long ...
and
semi-minor axis
In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both focus (geometry), foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major wikt: ...
of an
ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
.
In
spherical coordinate system
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
for which
, the general ellipsoid is defined as:
:
where
is the polar angle and
is the azimuthal angle.
When
, the ellipsoid is a sphere.
When
, the ellipsoid is a spheroid or ellipsoid of revolution. In particular, if
, it is an
oblate spheroid
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 axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circ ...
; if
, it is a
prolate spheroid
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 axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circu ...
.
Parameterization
The ellipsoid may be parameterized in several ways, which are simpler to express when the ellipsoid axes coincide with coordinate axes. A common choice is
:
where
:
These parameters may be interpreted as
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
, where is the polar angle and is the azimuth angle of the point of the ellipsoid.
Measuring from the center rather than a pole,
:
where
:
is the
reduced latitude
In geography, latitude is a coordinate that specifies the north–south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...
,
parametric latitude
In geography, latitude is a coordinate that specifies the north–south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pole ...
, or
eccentric anomaly In orbital mechanics, the eccentric anomaly is an angular parameter that defines the position of a body that is moving along an elliptic Kepler orbit. The eccentric anomaly is one of three angular parameters ("anomalies") that define a position alo ...
and is azimuth or longitude.
Measuring angles directly to the surface of the ellipsoid, not to the circumscribed sphere,
:
where
:
would be
geocentric latitude
In geography, latitude is a coordinate that specifies the north–south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pole ...
on the Earth, and is longitude. These are true spherical coordinates with the origin at the center of the ellipsoid.
In
geodesy
Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equivale ...
, the
geodetic latitude
Geodetic coordinates are a type of curvilinear orthogonal coordinate system used in geodesy based on a '' reference ellipsoid''.
They include geodetic latitude (north/south) , ''longitude'' (east/west) , and ellipsoidal height (also known as ...
is most commonly used, as the angle between the vertical and the equatorial plane, defined for a biaxial ellipsoid. For a more general triaxial ellipsoid, see
ellipsoidal latitude
The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an ''oblate ellipsoid'', a slightly flattened sphere. A ''geodesi ...
.
Volume
The
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
bounded by the ellipsoid is
:
In terms of the principal
diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for ...
s (where , , ), the volume is
:
.
This equation reduces to that of the volume of a sphere when all three elliptic radii are equal, and to that of an
oblate
In Christianity (especially in the Roman Catholic, Orthodox, Anglican and Methodist traditions), an oblate is a person who is specifically dedicated to God or to God's service.
Oblates are individuals, either laypersons or clergy, normally livi ...
or
prolate spheroid
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 axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circu ...
when two of them are equal.
The
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
of an ellipsoid is the volume of a
circumscribed
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
elliptic cylinder, and the volume of the circumscribed box. The
volumes
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The defi ...
of the
inscribed
{{unreferenced, date=August 2012
An inscribed triangle of a circle
In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figur ...
and circumscribed
boxes are respectively:
:
Surface area
The
surface area
The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc ...
of a general (triaxial) ellipsoid is
:
where
:
and where and are incomplete
elliptic integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...
s of the first and second kind respectively. The surface area of this general ellipsoid can also be expressed using the and
Carlson symmetric form In mathematics, the Carlson symmetric forms of elliptic integrals are a small canonical set of elliptic integrals to which all others may be reduced. They are a modern alternative to the Legendre forms. The Legendre forms may be expressed in terms ...
s of the elliptic integrals by simply substituting the above formula to the respective definitions:
:
Unlike the expression with and , the variant based on the Carlson symmetric integrals yields valid results for a sphere and only the axis must be the smallest, the order between the two larger axes, and can be arbitrary.
The surface area of an ellipsoid of revolution (or spheroid) may be expressed in terms of
elementary function
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponen ...
s:
:
or
:
or
:
and
:
which, as follows from basic trigonometric identities, are equivalent expressions (i.e. the formula for can be used to calculate the surface area of a prolate ellipsoid and vice versa). In both cases may again be identified as the
eccentricity
Eccentricity or eccentric may refer to:
* Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal"
Mathematics, science and technology Mathematics
* Off-center, in geometry
* Eccentricity (graph theory) of a v ...
of the ellipse formed by the cross section through the symmetry axis. (See
ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
). Derivations of these results may be found in standard sources, for example
Mathworld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Dig ...
.
Approximate formula
:
Here yields a relative error of at most 1.061%; a value of is optimal for nearly spherical ellipsoids, with a relative error of at most 1.178%.
In the "flat" limit of much smaller than and , the area is approximately , equivalent to .
Plane sections
![Ellipsoid-ebener-Schnitt](https://upload.wikimedia.org/wikipedia/commons/0/01/Ellipsoid-ebener-Schnitt.svg)
The intersection of a plane and a sphere is a circle (or is reduced to a single point, or is empty). Any ellipsoid is the image of the unit sphere under some affine transformation, and any plane is the image of some other plane under the same transformation. So, because affine transformations map circles to ellipses, the intersection of a plane with an ellipsoid is an ellipse or a single point, or is empty. Obviously, spheroids contain circles. This is also true, but less obvious, for triaxial ellipsoids (see
Circular section
In geometry, a circular section is a circle on a quadric surface (such as an ellipsoid or hyperboloid). It is a special plane section of the quadric, as this circle is the intersection with the quadric of the plane containing the circle.
Any plan ...
).
Determining the ellipse of a plane section
![Ellipso-eb-beisp](https://upload.wikimedia.org/wikipedia/commons/0/00/Ellipso-eb-beisp.svg)
Given: Ellipsoid and the plane with equation , which have an ellipse in common.
Wanted: Three vectors (center) and , (conjugate vectors), such that the ellipse can be represented by the parametric equation
:
(see
ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
).
![Ellipso-eb-ku](https://upload.wikimedia.org/wikipedia/commons/3/3c/Ellipso-eb-ku.svg)
Solution: The scaling transforms the ellipsoid onto the unit sphere and the given plane onto the plane with equation
:
Let be the
Hesse normal form
The Hesse normal form named after Otto Hesse, is an equation used in analytic geometry, and describes a line in \mathbb^2 or a plane in Euclidean space \mathbb^3 or a hyperplane in higher dimensions.John Vince: ''Geometry for Computer Graphics''. S ...
of the new plane and
:
its unit normal vector. Hence
:
is the ''center'' of the intersection circle and
:
its radius (see diagram).
Where (i.e. the plane is horizontal), let
:
Where , let
:
In any case, the vectors are orthogonal, parallel to the intersection plane and have length (radius of the circle). Hence the intersection circle can be described by the parametric equation
:
The reverse scaling (see above) transforms the unit sphere back to the ellipsoid and the vectors are mapped onto vectors , which were wanted for the parametric representation of the intersection ellipse.
How to find the vertices and semi-axes of the ellipse is described in
ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
.
Example: The diagrams show an ellipsoid with the semi-axes which is cut by the plane .
Pins-and-string construction
![Fokalks-ellipsoid-xyz](https://upload.wikimedia.org/wikipedia/commons/b/b3/Fokalks-ellipsoid-xyz.svg)
The pins-and-string construction of an ellipsoid is a transfer of the idea constructing an ellipse using two
pins and a string (see diagram).
A pins-and-string construction of an
ellipsoid of revolution
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 axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has c ...
is given by the pins-and-string construction of the rotated ellipse.
The construction of points of a ''triaxial ellipsoid'' is more complicated. First ideas are due to the Scottish physicist
J. C. Maxwell (1868). Main investigations and the extension to quadrics was done by the German mathematician O. Staude in 1882, 1886 and 1898. The description of the pins-and-string construction of ellipsoids and hyperboloids is contained in the book ''Geometry and the imagination'' written by
D. Hilbert & S. Vossen, too.
Steps of the construction
# Choose an ''ellipse'' and a ''hyperbola'' , which are a pair of
focal conics
In geometry, focal conics are a pair of curves consisting of
either
*an ellipse and a hyperbola, where the hyperbola is contained in a plane, which is orthogonal to the plane containing the ellipse. The vertices of the hyperbola are the foci of ...
:
with the vertices and foci of the ellipse
and a ''string'' (in diagram red) of length .
# Pin one end of the string to
vertex
Vertex, vertices or vertexes may refer to:
Science and technology Mathematics and computer science
*Vertex (geometry), a point where two or more curves, lines, or edges meet
*Vertex (computer graphics), a data structure that describes the position ...
and the other to focus . The string is kept tight at a point with positive - and -coordinates, such that the string runs from to behind the upper part of the hyperbola (see diagram) and is free to slide on the hyperbola. The part of the string from to runs and slides in front of the ellipse. The string runs through that point of the hyperbola, for which the distance over any hyperbola point is at a minimum. The analogous statement on the second part of the string and the ellipse has to be true, too.
# Then: is a point of the ellipsoid with equation
# The remaining points of the ellipsoid can be constructed by suitable changes of the string at the focal conics.
Semi-axes
Equations for the semi-axes of the generated ellipsoid can be derived by special choices for point :
:
The lower part of the diagram shows that and are the foci of the ellipse in the -plane, too. Hence, it is
confocal In geometry, confocal means having the same foci: confocal conic sections.
* For an optical cavity consisting of two mirrors, confocal means that they share their foci. If they are identical mirrors, their radius of curvature, ''R''mirror, equals ' ...
to the given ellipse and the length of the string is . Solving for yields ; furthermore .
From the upper diagram we see that and are the foci of the ellipse section of the ellipsoid in the -plane and that .
Converse
If, conversely, a triaxial ellipsoid is given by its equation, then from the equations in step 3 one can derive the parameters , , for a pins-and-string construction.
Confocal ellipsoids
If is an ellipsoid
confocal In geometry, confocal means having the same foci: confocal conic sections.
* For an optical cavity consisting of two mirrors, confocal means that they share their foci. If they are identical mirrors, their radius of curvature, ''R''mirror, equals ' ...
to with the squares of its semi-axes
:
then from the equations of
:
one finds, that the corresponding focal conics used for the pins-and-string construction have the same semi-axes as ellipsoid . Therefore (analogously to the foci of an ellipse) one considers the focal conics of a triaxial ellipsoid as the (infinite many) foci and calls them the focal curves of the ellipsoid.
The converse statement is true, too: if one chooses a second string of length and defines
:
then the equations
:
are valid, which means the two ellipsoids are confocal.
Limit case, ellipsoid of revolution
In case of (a
spheroid
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 axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has cir ...
) one gets and , which means that the focal ellipse degenerates to a line segment and the focal hyperbola collapses to two infinite line segments on the -axis. The ellipsoid is
rotationally symmetric around the -axis and
:
.
Properties of the focal hyperbola
![Ellipsoid-pk-zk](https://upload.wikimedia.org/wikipedia/commons/c/cf/Ellipsoid-pk-zk.svg)
; True curve
: If one views an ellipsoid from an external point of its focal hyperbola, than it seems to be a sphere, that is its apparent shape is a circle. Equivalently, the tangents of the ellipsoid containing point are the lines of a circular
cone
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines con ...
, whose axis of rotation is the
tangent line
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
of the hyperbola at . If one allows the center to disappear into infinity, one gets an
orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
parallel projection
In three-dimensional geometry, a parallel projection (or axonometric projection) is a projection of an object in three-dimensional space onto a fixed plane, known as the ''projection plane'' or '' image plane'', where the ''rays'', known as '' li ...
with the corresponding
asymptote
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
of the focal hyperbola as its direction. The ''true curve of shape'' (tangent points) on the ellipsoid is not a circle. The lower part of the diagram shows on the left a parallel projection of an ellipsoid (with semi-axes 60, 40, 30) along an asymptote and on the right a central projection with center and main point on the tangent of the hyperbola at point . ( is the foot of the perpendicular from onto the image plane.) For both projections the apparent shape is a circle. In the parallel case the image of the origin is the circle's center; in the central case main point is the center.
; Umbilical points
: The focal hyperbola intersects the ellipsoid at its four
umbilical point
In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are eq ...
s.
Property of the focal ellipse
The focal ellipse together with its inner part can be considered as the limit surface (an infinitely thin ellipsoid) of the
pencil
A pencil () is a writing or drawing implement with a solid pigment core in a protective casing that reduces the risk of core breakage, and keeps it from marking the user's hand.
Pencils create marks by physical abrasion, leaving a trail ...
of confocal ellipsoids determined by for . For the limit case one gets
:
In general position
As a quadric
If is a point and is a real, symmetric,
positive-definite matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a co ...
, then the set of points that satisfy the equation
:
is an ellipsoid centered at . The
eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of are the principal axes of the ellipsoid, and the
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of are the reciprocals of the squares of the semi-axes: , and .
An invertible
linear transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
applied to a sphere produces an ellipsoid, which can be brought into the above standard form by a suitable
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
, a consequence of the
polar decomposition
In mathematics, the polar decomposition of a square real or complex matrix A is a factorization of the form A = U P, where U is an orthogonal matrix and P is a positive semi-definite symmetric matrix (U is a unitary matrix and P is a positive semi ...
(also, see
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix (mathematics), matrix can be Diagonalizable matrix, diagonalized (that is, represented as a diagonal matrix i ...
). If the linear transformation is represented by a
symmetric 3 × 3 matrix, then the eigenvectors of the matrix are orthogonal (due to the
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix (mathematics), matrix can be Diagonalizable matrix, diagonalized (that is, represented as a diagonal matrix i ...
) and represent the directions of the axes of the ellipsoid; the lengths of the semi-axes are computed from the eigenvalues. The
singular value decomposition
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is related ...
and
polar decomposition
In mathematics, the polar decomposition of a square real or complex matrix A is a factorization of the form A = U P, where U is an orthogonal matrix and P is a positive semi-definite symmetric matrix (U is a unitary matrix and P is a positive semi ...
are matrix decompositions closely related to these geometric observations.
Parametric representation
![Ellipsoid-affin](https://upload.wikimedia.org/wikipedia/commons/0/0e/Ellipsoid-affin.svg)
The key to a parametric representation of an ellipsoid in general position is the alternative definition:
: ''An ellipsoid is an affine image of the unit sphere.''
An
affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generally, ...
can be represented by a translation with a vector and a regular 3 × 3 matrix :
:
where are the column vectors of matrix .
A parametric representation of an ellipsoid in general position can be obtained by the parametric representation of a unit sphere (see above) and an affine transformation:
:
.
If the vectors form an orthogonal system, the six points with vectors are the vertices of the ellipsoid and are the semi-principal axes.
A surface normal vector at point is
:
For any ellipsoid there exists an
implicit representation . If for simplicity the center of the ellipsoid is the origin, , the following equation describes the ellipsoid above:
:
Applications
The ellipsoidal shape finds many practical applications:
;
Geodesy
Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equivale ...
*
Earth ellipsoid
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, and the geosciences. Various different ellipsoids have been used as approximations ...
, a mathematical figure approximating the shape of the
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
.
*
Reference ellipsoid
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, and the geosciences. Various different ellipsoids have been used as approximations ...
, a mathematical figure approximating the shape of
planet
A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
ary bodies in general.
;
Mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
*
Poinsot's ellipsoid
In classical mechanics, Poinsot's construction (after Louis Poinsot) is a geometrical method for visualizing the torque-free motion of a rotating rigid body, that is, the motion of a rigid body on which no external forces are acting. This motion ha ...
, a geometrical method for visualizing the
torque-free motion of a rotating
rigid body
In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force ...
.
*
Lamé's stress ellipsoid
Lamé's stress ellipsoid is an alternative to Mohr's circle for the graphical representation of the stress state at a point. The surface of the ellipsoid represents the locus of the endpoints of all stress vectors acting on all planes passing thro ...
, an alternative to
Mohr's circle
Mohr's circle is a two-dimensional graphical representation of the transformation law for the Cauchy stress tensor.
Mohr's circle is often used in calculations relating to mechanical engineering for materials' strength, geotechnical engineer ...
for the graphical representation of the
stress
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
state at a point.
*
Manipulability ellipsoid, used to describe a robot's freedom of motion.
*
Jacobi ellipsoid
A Jacobi ellipsoid is a triaxial (i.e. scalene) ellipsoid under hydrostatic equilibrium which arises when a self-gravitating fluid body of uniform density rotates with a constant angular velocity. It is named after the German mathematician Carl Gu ...
, a triaxial ellipsoid formed by a rotating fluid
;
Crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ...
*
Index ellipsoid In crystal optics, the index ellipsoid (also known as the ''optical indicatrix'' or sometimes as the ''dielectric ellipsoid'') is a geometric construction which concisely represents the refractive indices and associated polariz ...
, a diagram of an ellipsoid that depicts the orientation and relative magnitude of
refractive indices
In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
The refractive index determines how much the path of light is bent, or ...
in a
crystal
A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...
.
*
Thermal ellipsoid
Thermal ellipsoids, more formally termed atomic displacement parameters or anisotropic displacement parameters, are ellipsoids used in crystallography to indicate the magnitudes and directions of the thermal vibration of atoms in crystal structu ...
, ellipsoids used in crystallography to indicate the magnitudes and directions of the
thermal vibration
The term "thermal energy" is used loosely in various contexts in physics and engineering. It can refer to several different well-defined physical concepts. These include the internal energy or enthalpy of a body of matter and radiation; heat, de ...
of atoms in
crystal structure
In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystal, crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric pat ...
s.
;Lighting
*
Ellipsoidal reflector floodlight
In stage lighting, an ellipsoidal reflector floodlight (sometimes known by the acronym ERF which is often pronounced "erf"), better known as a scoop, is a large, simple lighting fixture with a dome-like reflector, large high-wattage lamp and no ...
*
Ellipsoidal reflector spotlight
Ellipsoidal reflector spot (abbreviated to ERS, or colloquially ellipsoidal or ellipse) is the name for a type of stage lighting instrument, named for the ellipsoidal reflector used to collect and direct the light through a barrel that co ...
;Medicine
* Measurements obtained from
MRI
Magnetic resonance imaging (MRI) is a medical imaging technique used in radiology to form pictures of the anatomy and the physiological processes of the body. MRI scanners use strong magnetic fields, magnetic field gradients, and radio waves ...
imaging of the
prostate
The prostate is both an Male accessory gland, accessory gland of the male reproductive system and a muscle-driven mechanical switch between urination and ejaculation. It is found only in some mammals. It differs between species anatomically, ...
can be used to determine the volume of the gland using the approximation (where 0.52 is an approximation for )
Dynamical properties
The
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
of an ellipsoid of uniform
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
is
:
The
moments of inertia
The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular accelera ...
of an ellipsoid of uniform density are
:
For these moments of inertia reduce to those for a sphere of uniform density.
![2003EL61art](https://upload.wikimedia.org/wikipedia/commons/9/90/2003EL61art.jpg)
Ellipsoids and
cuboid
In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a cub ...
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
The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...
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 Earth, which is merely
oblate
In Christianity (especially in the Roman Catholic, Orthodox, Anglican and Methodist traditions), an oblate is a person who is specifically dedicated to God or to God's service.
Oblates are individuals, either laypersons or clergy, normally livi ...
); in addition, because of
tidal locking
Tidal locking between a pair of co-orbiting astronomical bodies occurs when one of the objects reaches a state where there is no longer any net change in its rotation rate over the course of a complete orbit. In the case where a tidally locked b ...
, moons in
synchronous orbit A synchronous orbit is an orbit in which an orbiting body (usually a satellite) has a period equal to the average rotational period of the body being orbited (usually a planet), and in the same direction of rotation as that body.
Simplified meanin ...
such as
Mimas orbit with their major axis aligned radially to their planet.
A spinning body of homogeneous self-gravitating fluid will assume the form of either a
Maclaurin spheroid (oblate spheroid) or
Jacobi ellipsoid
A Jacobi ellipsoid is a triaxial (i.e. scalene) ellipsoid under hydrostatic equilibrium which arises when a self-gravitating fluid body of uniform density rotates with a constant angular velocity. It is named after the German mathematician Carl Gu ...
(scalene ellipsoid) when in
hydrostatic equilibrium
In fluid mechanics, hydrostatic equilibrium (hydrostatic balance, hydrostasy) is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force. In the planetary ...
, and for moderate rates of rotation. At faster rotations, non-ellipsoidal
piriform or
oviform
An oval () is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas (projective geometry, technical drawing, etc.) it is given a more precise definition, which may include either one o ...
shapes can be expected, but these are not stable.
Fluid dynamics
The ellipsoid is the most general shape for which it has been possible to calculate the
creeping flow
Stokes flow (named after George Gabriel Stokes), also named creeping flow or creeping motion,Kim, S. & Karrila, S. J. (2005) ''Microhydrodynamics: Principles and Selected Applications'', Dover. . is a type of fluid flow where advection, advec ...
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
A microorganism, or microbe,, ''mikros'', "small") and ''organism'' from the el, ὀργανισμός, ''organismós'', "organism"). It is usually written as a single word but is sometimes hyphenated (''micro-organism''), especially in olde ...
.
In probability and statistics
The
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. Intuitively, in the simplified two and three dimensional case, the joint ...
s, which generalize the
multivariate normal distribution
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One d ...
and are used in
finance
Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of fina ...
, can be defined in terms of their
density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
s. When they exist, the density functions have the structure:
:
where is a scale factor, is an -dimensional
random row vector with median vector (which is also the mean vector if the latter exists), is a
positive definite matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a co ...
which is proportional to the
covariance matrix
In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...
if the latter exists, and is a function mapping from the non-negative reals to the non-negative reals giving a finite area under the curve.
[Frahm, G., Junker, M., & Szimayer, A. (2003). Elliptical copulas: applicability and limitations. Statistics & Probability Letters, 63(3), 275–286.] The multivariate normal distribution is the special case in which for quadratic form .
Thus the density function is a scalar-to-scalar transformation of a quadric expression. Moreover, the equation for any
iso-density surface states that the quadric expression equals some constant specific to that value of the density, and the iso-density surface is an ellipsoid.
In higher dimensions
A hyperellipsoid, or ellipsoid of dimension
in a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
of dimension
, is a
quadric hypersurface defined by a polynomial of degree two that has a
homogeneous part of degree two which is a
positive definite quadratic form
Positive is a property of positivity and may refer to:
Mathematics and science
* Positive formula, a logical formula not containing negation
* Positive number, a number that is greater than 0
* Plus sign, the sign "+" used to indicate a posit ...
.
One can also define a hyperellipsoid as the image of a sphere under an invertible
affine transformation. The spectral theorem can again be used to obtain a standard equation of the form
:
The volume of an -dimensional ''hyperellipsoid'' can be obtained by replacing by the product of the semi-axes in the formula for the
volume of a hypersphere:
:
(where is the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
).
See also
*
Ellipsoidal dome
An ellipsoidal dome is a dome (also see geodesic dome), which has a bottom cross-section which is a circle, but has a cupola whose curve is an ellipse.
There are two types of ellipsoidal domes: ''prolate ellipsoidal domes'' and ''oblate ellipso ...
*
Ellipsoid method
In mathematical optimization, the ellipsoid method is an iterative method for convex optimization, minimizing convex functions. When specialized to solving feasible linear optimization problems with rational data, the ellipsoid method is an algor ...
*
Ellipsoidal coordinates
Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system (\lambda, \mu, \nu) that generalizes the two-dimensional elliptic coordinate system. Unlike most three-dimensional orthogonal coordinate systems that feature quadratic ...
*
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. Intuitively, in the simplified two and three dimensional case, the joint ...
, in statistics
*
Flattening
Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution (spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening is ...
, also called ''
ellipticity
Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution (spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening is ...
'' and ''
oblateness
Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution (spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening is ...
'', is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution (spheroid), respectively.
*
Focaloid
In geometry, a focaloid is a shell bounded by two concentric, confocal ellipses (in 2D) or ellipsoids (in 3D). When the thickness of the shell becomes negligible, it is called a thin focaloid.
Mathematical definition (3D)
If one boundary ...
, a shell bounded by two concentric, confocal ellipsoids
*
Geodesics on an ellipsoid
The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an ''oblate ellipsoid'', a slightly flattened sphere. A ''geodesi ...
*
Geodetic datum
A geodetic datum or geodetic system (also: geodetic reference datum, geodetic reference system, or geodetic reference frame) is a global datum reference or reference frame for precisely representing the position of locations on Earth or other plan ...
, the gravitational Earth modeled by a best-fitted ellipsoid
*
Homoeoid
A homoeoid is a shell (a bounded region) bounded by two concentric, similar ellipses (in 2D) or ellipsoids (in 3D).
When the thickness of the shell becomes negligible, it is called a thin homoeoid. The name homoeoid was coined by Lord Kelvin and ...
, a shell bounded by two concentric similar ellipsoids
*
List of surfaces
This is a list of surfaces, by Wikipedia page.
''See also List of algebraic surfaces, List of curves, Riemann surface.''
Minimal surfaces
* Catalan's minimal surface
* Costa's minimal surface
* Catenoid
* Enneper surface
* Gyroid
* Helicoid
* Lid ...
*
Superellipsoid
In mathematics, a superellipsoid (or super-ellipsoid) is a solid whose horizontal sections are superellipses (Lamé curves) with the same exponent ''r'', and whose vertical sections through the center are superellipses with the same exponent ''t ...
Notes
References
*
External links
{{Commons category, Ellipsoids
*
Ellipsoid by Jeff Bryant,
Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
, 2007.
Ellipsoidan
MathWorld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Dig ...
.
Geometric shapes
Surfaces
Quadrics
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