NationMaster - Encyclopedia: Quadric surface

This is an article about quadric in mathematics, to see the computing company go to Quadrics. This is an article about the computing company, for use in mathematics, see quadric. ...

In mathematics a quadric, or quadric surface, is any D-dimensional hypersurface defined as the locus of zeros of a quadratic polynomial. In coordinates ${x_0, x_1, x_2, ldots, x_D}$ , the general quadric is defined by the algebraic equation ^[1] Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... In mathematics, a hypersurface is some kind of submanifold. ... In mathematics, a locus (Latin for place, plural loci) is a collection of points which share a common property. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... f(x) = x2 - x - 2 In mathematics, a quadratic function is a polynomial function of the form , where a is nonzero. ...

$sum_{i,j=0}^D Q_{i,j} x_i x_j + sum_{i=0}^D P_i x_i + R = 0$

where Q is a D+1 dimensional matrix and P is a D+1 dimensional vector and R a constant. The values Q, P and R are often taken to be real numbers or complex numbers, but in fact, a quadric may be defined over any field. In general, the locus of zeros of a set of polynomials is known as an algebraic variety, and is studied in the branch of algebraic geometry. A quadric is thus an example of an algebraic variety. Every projective variety can be shown to be isomorphic to the intersection of a set of quadrics. In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = âˆ’1. ... This article presents the essential definitions. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... This article is about algebraic varieties. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...

The normalized equation for a three-dimensional (D=3) quadric centred at the origin (0,0,0) is:

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

Via translations and rotations every quadric can be transformed to one of several "normalized" forms. In three-dimensional Euclidean space there are 16 such normalized forms, and the most interesting, the nondegenerate forms are given below. The remaining forms are called degenerate forms and include planes, lines, points or even no points at all. ^[2] Two intersecting planes in R3 In mathematics, a plane is a fundamental two-dimensional object. ... A line, or straight line, can be described as an (infinitely) thin, (infinitely) long, perfectly straight curve (the term curve in mathematics includes straight curves). In Euclidean geometry, exactly one line can be found that passes through any two points. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ...

Ellipsoid Ellipsoid File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...	Elliptic Paraboloid Elliptic Paraboloid File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...	Hyperbolic Paraboloid Hyperbolic Paraboloid File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...	Hyperboloid of One Sheet Hyperboloid of One Sheet File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...
Hyperboloid of Two Sheets Hyperboloid of Two Sheets File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...	Cone Cone File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...	Elliptic Cylinder Elliptic Cylinder File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...	Hyperbolic Cylinder Hyperbolic Cylinder File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...
Parabolic Cylinder Parabolic Cylinder File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

ellipsoid	${x^2 over a^2} + {y^2 over b^2} + {z^2 over c^2} = 1 ,$
spheroid (special case of ellipsoid)	${x^2 over a^2} + {y^2 over a^2} + {z^2 over b^2} = 1 ,$
sphere (special case of spheroid)	${x^2 over a^2} + {y^2 over a^2} + {z^2 over a^2} = 1 ,$
elliptic paraboloid	${x^2 over a^2} + {y^2 over b^2} - z = 0 ,$
circular paraboloid	${x^2 over a^2} + {y^2 over a^2} - z = 0 ,$
hyperbolic paraboloid	${x^2 over a^2} - {y^2 over b^2} - z = 0 ,$
hyperboloid of one sheet	${x^2 over a^2} + {y^2 over b^2} - {z^2 over c^2} = 1 ,$
hyperboloid of two sheets	${x^2 over a^2} - {y^2 over b^2} - {z^2 over c^2} = 1 ,$
cone	${x^2 over a^2} + {y^2 over b^2} - {z^2 over c^2} = 0 ,$
elliptic cylinder	${x^2 over a^2} + {y^2 over b^2} = 1 ,$
circular cylinder	${x^2 over a^2} + {y^2 over a^2} = 1 ,$
hyperbolic cylinder	${x^2 over a^2} - {y^2 over b^2} = 1 ,$
parabolic cylinder	$x^2 + 2y = 0 ,$

In real projective space, the ellipsoid, the elliptic paraboloid and the hyperboloid of two sheets are equivalent to each other up to a projective transformation; the two hyperbolic paraboloids are not different from each other (these are ruled surfaces); the cone and the cylinder are not different from each other (these are "degenerate" quadrics, since their Gaussian curvature is zero). 3D rendering of an ellipsoid In mathematics, an ellipsoid is a type of quadric that is a higher dimensional analogue of an ellipse. ... In mathematics, a spheroid is a quadric surface in three dimensions obtained by rotating an ellipse about one of its principal axes. ... A sphere (< Greek ÏƒÏ†Î±Î¯ÏÎ±) is a perfectly symmetrical geometrical object. ... Paraboloid of revolution Hyperbolic paraboloid In mathematics, a paraboloid is a quadric, a type of surface in three dimensions, described by the equation: (elliptic paraboloid), or (hyperbolic paraboloid). ... Paraboloid of revolution Hyperbolic paraboloid In mathematics, a paraboloid is a quadric, a type of surface in three dimensions, described by the equation: (elliptic paraboloid), or (hyperbolic paraboloid). ... Paraboloid of revolution Hyperbolic paraboloid In mathematics, a paraboloid is a quadric, a type of surface in three dimensions, described by the equation: (elliptic paraboloid), or (hyperbolic paraboloid). ... Hyperboloid of one sheet Hyperboloid of two sheets In mathematics, a hyperboloid is a quadric, a type of surface in three dimensions, described by the equation: (hyperboloid of one sheet), or (hyperboloid of two sheets) If, and only if, , it is a hyperboloid of revolution. ... Hyperboloid of one sheet Hyperboloid of two sheets In mathematics, a hyperboloid is a quadric, a type of surface in three dimensions, described by the equation: (hyperboloid of one sheet), or (hyperboloid of two sheets) If, and only if, , it is a hyperboloid of revolution. ... In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point â€” the apex or vertex â€” and any point of some fixed space curve â€” the directrix â€” that does not contain the apex. ... A right circular cylinder In mathematics, a cylinder is a quadric, i. ... A right circular cylinder In mathematics, a cylinder is a quadric, i. ... A right circular cylinder In mathematics, a cylinder is a quadric, i. ... A right circular cylinder In mathematics, a cylinder is a quadric, i. ... In mathematics, real projective space, or RPn is the projective space of lines in Rn+1. ... Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... A projective transformation is a transformation used in projective geometry: it is the composition of a pair of perspective projections. ... In geometry, a surface is ruled if through every point of there is a straight line that lies on . ... Curvature is the amount by which a geometric object deviates from being flat. ...

In complex projective space all of the nondegenerate quadrics become indistinguishable from each other. In mathematics, complex projective space, or CPn, is the projective space of (complex) lines in Cn+1. ...

References

^ [1], Quadrics in Geometry Formulas and Facts by Silvio Levy, excerpted from 30th Edition of the CRC Standard Mathematical Tables and Formulas (CRC Press).
^ Stewart Venit and Wayne Bishop, Elementary Linear Algebra (fourth edition), International Thompson Publishing, 1996.

External links

Quadric on MathWorld

Categories: Geometry | Surfaces | Quadrics

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