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Index Ellipsoid

An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. [1]

82 relations: Affine transformation, Bounded set, Cartesian coordinate system, Circular section, Circumscribed circle, Contour line, Covariance matrix, Cross section (geometry), Cuboid, Cylinder, Earth ellipsoid, Eccentricity (mathematics), Eigenvalues and eigenvectors, Elementary function, Ellipse, Ellipsoid method, Ellipsoidal coordinates, Ellipsoidal reflector spotlight, Elliptic integral, Elliptical distribution, Euclidean vector, Finance, Flattening, Focaloid, Hesse normal form, Homoeoid, Hydrostatic equilibrium, Hyperboloid, Implicit surface, Index ellipsoid, Inscribed figure, Jacobi ellipsoid, Lamé's stress ellipsoid, Line segment, Linear map, Maclaurin spheroid, Magnetic resonance imaging, Manipulability ellipsoid, Mass, MathWorld, Microorganism, Mimas (moon), Moment of inertia, Multivariate normal distribution, Multivariate random variable, Oval, Paraboloid, Perpendicular, Poinsot's ellipsoid, Point reflection, ..., Polar decomposition, Polynomial, Positive-definite matrix, Probability density function, Prostate, Quadric, Real number, Reference ellipsoid, Rigid body, Rotation, Rotational symmetry, Scaling (geometry), Scoop (theater), Semi-major and semi-minor axes, Singular-value decomposition, Spectral theorem, Sphere, Spherical coordinate system, Spheroid, Stokes flow, Superellipsoid, Surface (mathematics), Surface area, Surface of revolution, Symmetric matrix, Synchronous orbit, Thermal ellipsoid, Tidal locking, Volume, Volume of an n-ball, Wolfram Demonstrations Project, Zero of a function. Expand index (32 more) »

Affine transformation

In geometry, an affine transformation, affine mapBerger, Marcel (1987), p. 38.

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Bounded set

In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size.

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Cartesian coordinate system

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.

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Circular section

In geometry a circular section is a circle on a quadric surface (such as an ellipsoid or hyperboloid).

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Circumscribed circle

In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon.

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Contour line

A contour line (also isocline, isopleth, isarithm, or equipotential curve) of a function of two variables is a curve along which the function has a constant value, so that the curve joins points of equal value.

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Covariance matrix

In probability theory and statistics, a covariance matrix (also known as dispersion matrix or variance–covariance matrix) is a matrix whose element in the i, j position is the covariance between the i-th and j-th elements of a random vector.

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Cross section (geometry)

In geometry and science, a cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional spaces.

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In geometry, a cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube.

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A cylinder (from Greek κύλινδρος – kulindros, "roller, tumbler"), has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes.

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Earth ellipsoid

An Earth ellipsoid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy, and the geosciences.

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Eccentricity (mathematics)

In mathematics, the eccentricity, denoted e or \varepsilon, is a parameter associated with every conic section.

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Eigenvalues and eigenvectors

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.

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Elementary function

In mathematics, an elementary function is a function of one variable which is the composition of a finite number of arithmetic operations, exponentials, logarithms, constants, and solutions of algebraic equations (a generalization of ''n''th roots).

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In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve.

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Ellipsoid method

In mathematical optimization, the ellipsoid method is an iterative method for minimizing convex functions.

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Ellipsoidal coordinates

Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system (\lambda, \mu, \nu) that generalizes the two-dimensional elliptic coordinate system.

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Ellipsoidal reflector spotlight

Ellipsoidal reflector light (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 contains a lens or lens train.

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Elliptic integral

In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse.

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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.

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Euclidean vector

In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction.

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Finance is a field that is concerned with the allocation (investment) of assets and liabilities (known as elements of the balance statement) over space and time, often under conditions of risk or uncertainty.

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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.

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In geometry, a focaloid is a shell bounded by two concentric, confocal ellipses (in 2D) or ellipsoids (in 3D).

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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.

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A homoeoid is a shell (a bounded region) bounded by two concentric, similar ellipses (in 2D) or ellipsoids (in 3D).

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Hydrostatic equilibrium

In fluid mechanics, a fluid is said to be in hydrostatic equilibrium or hydrostatic balance when it is at rest, or when the flow velocity at each point is constant over time.

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In geometry, a hyperboloid of revolution, sometimes called circular hyperboloid, is a surface that may be generated by rotating a hyperbola around one of its principal axes.

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Implicit surface

In mathematics an implicit surface is a surface in Euclidean space defined by an equation An implicit surface is the set of zeros of a function of three variables.

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Index ellipsoid

In optics, an index ellipsoid is a diagram of an ellipsoid that depicts the orientation and relative magnitude of refractive indices in a crystal.

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Inscribed figure

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.

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Jacobi ellipsoid

A Jacobi ellipsoid is a triaxial (i.e. scalene) ellipsoid under equilibrium which arises when a self-gravitating fluid body of uniform density rotates with a constant angular velocity.

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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.

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Line segment

In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints.

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Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

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Maclaurin spheroid

A Maclaurin spheroid is an oblate spheroid which arises when a self-gravitating fluid body of uniform density rotates with a constant angular velocity.

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Magnetic resonance imaging

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 in both health and disease.

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Manipulability ellipsoid

In robotics, the manipulability ellipsoid is the geometric interpretation of the scaled eigenvectors resulting from the singular value decomposition of the jacobian that describes a robot's motion.

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Mass is both a property of a physical body and a measure of its resistance to acceleration (a change in its state of motion) when a net force is applied.

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MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein.

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A microorganism, or microbe, is a microscopic organism, which may exist in its single-celled form or in a colony of cells. The possible existence of unseen microbial life was suspected from ancient times, such as in Jain scriptures from 6th century BC India and the 1st century BC book On Agriculture by Marcus Terentius Varro. Microbiology, the scientific study of microorganisms, began with their observation under the microscope in the 1670s by Antonie van Leeuwenhoek. In the 1850s, Louis Pasteur found that microorganisms caused food spoilage, debunking the theory of spontaneous generation. In the 1880s Robert Koch discovered that microorganisms caused the diseases tuberculosis, cholera and anthrax. Microorganisms include all unicellular organisms and so are extremely diverse. Of the three domains of life identified by Carl Woese, all of the Archaea and Bacteria are microorganisms. These were previously grouped together in the two domain system as Prokaryotes, the other being the eukaryotes. The third domain Eukaryota includes all multicellular organisms and many unicellular protists and protozoans. Some protists are related to animals and some to green plants. Many of the multicellular organisms are microscopic, namely micro-animals, some fungi and some algae, but these are not discussed here. They live in almost every habitat from the poles to the equator, deserts, geysers, rocks and the deep sea. Some are adapted to extremes such as very hot or very cold conditions, others to high pressure and a few such as Deinococcus radiodurans to high radiation environments. Microorganisms also make up the microbiota found in and on all multicellular organisms. A December 2017 report stated that 3.45 billion year old Australian rocks once contained microorganisms, the earliest direct evidence of life on Earth. Microbes are important in human culture and health in many ways, serving to ferment foods, treat sewage, produce fuel, enzymes and other bioactive compounds. They are essential tools in biology as model organisms and have been put to use in biological warfare and bioterrorism. They are a vital component of fertile soils. In the human body microorganisms make up the human microbiota including the essential gut flora. They are the pathogens responsible for many infectious diseases and as such are the target of hygiene measures.

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Mimas (moon)

Mimas, also designated Saturn I, is a moon of Saturn which was discovered in 1789 by William Herschel.

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Moment of inertia

The moment of inertia, otherwise known as the angular mass or rotational inertia, of a rigid body is a tensor that determines the torque needed for a desired angular acceleration about a rotational axis; similar to how mass determines the force needed for a desired acceleration.

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Multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution or multivariate Gaussian distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions.

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Multivariate random variable

In probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value.

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An oval (from Latin ovum, "egg") is a closed curve in a plane which "loosely" resembles the outline of an egg.

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In geometry, a paraboloid is a quadric surface that has (exactly) one axis of symmetry and no center of symmetry.

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In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees).

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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.

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Point reflection

In geometry, a point reflection or inversion in a point (or inversion through a point, or central inversion) is a type of isometry of Euclidean space.

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Polar decomposition

In mathematics, particularly in linear algebra and functional analysis, the polar decomposition of a matrix or linear operator is a factorization analogous to the polar form of a nonzero complex number z as z.

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In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

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Positive-definite matrix

In linear algebra, a symmetric real matrix M is said to be positive definite if the scalar z^Mz is strictly positive for every non-zero column vector z of n real numbers.

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Probability 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 be interpreted as providing a relative likelihood that the value of the random variable would equal that sample.

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The prostate (from Ancient Greek προστάτης, prostates, literally "one who stands before", "protector", "guardian") is a compound tubuloalveolar exocrine gland of the male reproductive system in most mammals.

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In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas).

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Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

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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.

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Rigid body

In physics, a rigid body is a solid body in which deformation is zero or so small it can be neglected.

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A rotation is a circular movement of an object around a center (or point) of rotation.

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Rotational symmetry

Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn.

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Scaling (geometry)

In Euclidean geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions.

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Scoop (theater)

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 lens.

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Semi-major and semi-minor axes

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 widest points of the perimeter.

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Singular-value decomposition

In linear algebra, the singular-value decomposition (SVD) is a factorization of a real or complex matrix.

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Spectral theorem

In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis).

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A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").

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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 measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane.

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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.

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Stokes 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.

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In mathematics, a super-ellipsoid or superellipsoid is a solid whose horizontal sections are super-ellipses (Lamé curves) with the same exponent r, and whose vertical sections through the center are super-ellipses with the same exponent t. Super-ellipsoids as computer graphics primitives were popularized by Alan H. Barr (who used the name "superquadrics" to refer to both superellipsoids and supertoroids).

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Surface (mathematics)

In mathematics, a surface is a generalization of a plane which needs not be flat, that is, the curvature is not necessarily zero.

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Surface area

The surface area of a solid object is a measure of the total area that the surface of the object occupies.

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Surface of revolution

A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation.

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Symmetric matrix

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose.

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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.

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Thermal ellipsoid

Thermal ellipsoids, more formally termed atomic displacement parameters, are ellipsoids used in crystallography to indicate the magnitudes and directions of the thermal vibration of atoms in crystal structures.

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Tidal locking

Tidal locking (also called gravitational locking or captured rotation) occurs when the long-term interaction between a pair of co-orbiting astronomical bodies drives the rotation rate of at least one of them into the state where there is no more net transfer of angular momentum between this body (e.g. a planet) and its orbit around the second body (e.g. a star); this condition of "no net transfer" must be satisfied over the course of one orbit around the second body.

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Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains.

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Volume of an n-ball

In geometry, a ball is a region in space comprising all points within a fixed distance from a given point; that is, it is the region enclosed by a sphere or hypersphere.

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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.

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Zero of a function

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, x is a solution of the equation f(x).

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Elipsoid, Ellipsoid (geometry), Ellipsoidal, Ellipsoids, Hyperellipsoid, Scalene ellipsoid, Semi-principal axis, Tri-axial ellipsoid, Triaxial ellipsoid.


[1] https://en.wikipedia.org/wiki/Ellipsoid

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