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In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface
Hypersurface

In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface....
 defined as the locus
Locus (mathematics)

In mathematics, a locus is a collection of point which share a property. The term locus is usually used of a condition which defines a continuous figure or figures, that is, a curve....
 of zeros
Root (mathematics)

In mathematics, a root of a complex-valued Function is a member of the Domain of such that vanishes at , that is,In other words, a "root" of a function is a value for that produces a result of zero ....
 of a quadratic polynomial
Quadratic polynomial

In mathematics, a quadratic polynomial or quadratic is a polynomial of degree of a polynomial two. A quadratic polynomial may involve a single variable x, or multiple variables such as x, y, and z....
. In coordinates , the general quadric is defined by the algebraic equation
Algebraic equation

In mathematics, an algebraic equation over a given Field is an equation of the formwhere P and Q are polynomials over that field. For example...
 

where Q is a (D + 1)×(D + 1) matrix
Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
 and P is a (D + 1)-dimensional vector and R a constant. The values Q, P and R are often taken to be real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s or complex number
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
s, but in fact, a quadric may be defined over any ring
Ring (mathematics)

In mathematics, a ring is a type of algebraic structure. There is some variation among mathematicians as to exactly what properties a ring is required to have, as described in detail below....
.






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In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface
Hypersurface

In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface....
 defined as the locus
Locus (mathematics)

In mathematics, a locus is a collection of point which share a property. The term locus is usually used of a condition which defines a continuous figure or figures, that is, a curve....
 of zeros
Root (mathematics)

In mathematics, a root of a complex-valued Function is a member of the Domain of such that vanishes at , that is,In other words, a "root" of a function is a value for that produces a result of zero ....
 of a quadratic polynomial
Quadratic polynomial

In mathematics, a quadratic polynomial or quadratic is a polynomial of degree of a polynomial two. A quadratic polynomial may involve a single variable x, or multiple variables such as x, y, and z....
. In coordinates , the general quadric is defined by the algebraic equation
Algebraic equation

In mathematics, an algebraic equation over a given Field is an equation of the formwhere P and Q are polynomials over that field. For example...
 

where Q is a (D + 1)×(D + 1) matrix
Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
 and P is a (D + 1)-dimensional vector and R a constant. The values Q, P and R are often taken to be real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s or complex number
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
s, but in fact, a quadric may be defined over any ring
Ring (mathematics)

In mathematics, a ring is a type of algebraic structure. There is some variation among mathematicians as to exactly what properties a ring is required to have, as described in detail below....
. In general, the locus of zeros of a set of polynomial
Polynomial

In mathematics, a polynomial is an expression constructed from variables and constants, using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents....
s is known as an algebraic variety
Algebraic variety

In mathematics, an algebraic variety is essentially a set of points where a polynomial or set of polynomials attain a value of zero. Algebraic varieties are one of the central objects of study in classical algebraic geometry....
, and is studied in the branch of algebraic geometry
Algebraic geometry

Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry....
.

A quadric is thus an example of an algebraic variety. For the projective theory see quadric (projective geometry)
Quadric (projective geometry)

In projective geometry a quadric is the set of points of a projective space where a certain quadratic form on the homogeneous coordinates becomes zero....
.

Low-Dimensional Real Quadrics


The one-dimensional (D=1) quadrics are the same as conic sections
Conic section

File:Conic sections with plane.svgIn mathematics, a conic section is a curve obtained by intersecting a cone with a plane . A conic section is therefore a restriction of a quadric surface to the plane ....
.

The normalized equation for a two-dimensional (D=2) quadric in three-dimensional space centred at the origin (0,0,0) is:

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
Degeneracy (mathematics)

In mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class....
 forms are given below. The remaining forms are called degenerate
Degeneracy (mathematics)

In mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class....
 forms and include plane
Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
s, line
Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
s, points or even no points at all.

ellipsoid
Ellipsoid

An ellipsoid is a type of Quadric that is a higher dimensional analogue of an ellipse. The equation of a standard axis-aligned ellipsoid body in an xyz-Cartesian coordinate system is...
    spheroid
Spheroid

A spheroid 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....
 (special case of ellipsoid)  
       sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
 (special case of spheroid)
elliptic 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....
    circular 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....
 (special case of elliptic paraboloid)
hyperbolic 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
Hyperboloid

In mathematics, a hyperboloid is a quadric, a type of surface in three dimensions, described by the equation  hyperboloid of one sheet,...
 of one sheet
hyperboloid
Hyperboloid

In mathematics, a hyperboloid is a quadric, a type of surface in three dimensions, described by the equation  hyperboloid of one sheet,...
 of two sheets
cone
Conical surface

In geometry, a conical surface is the unbounded surface formed by the union of all the straight line that pass through a fixed point — the apex or vertex — and any point of some fixed space curve — the directrix — that does not contain the apex....
elliptic cylinder
Cylinder (geometry)

A cylinder is one of the most curvilinear basic geometric shapes: the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder....
    circular cylinder
Cylinder (geometry)

A cylinder is one of the most curvilinear basic geometric shapes: the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder....
 (special case of elliptic cylinder)
hyperbolic cylinder
Cylinder (geometry)

A cylinder is one of the most curvilinear basic geometric shapes: the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder....
parabolic cylinder
Cylinder (geometry)

A cylinder is one of the most curvilinear basic geometric shapes: the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder....


A special fact about low-dimensional quadrics is that every real quadric (in any dimension) can be obtained starting from a conic
Conic section

File:Conic sections with plane.svgIn mathematics, a conic section is a curve obtained by intersecting a cone with a plane . A conic section is therefore a restriction of a quadric surface to the plane ....
 or the hyperbolic paraboloid by applying revolutions
Surface of revolution

A surface of revolution is a surface created by rotating a curve lying on some plane around a straight line that lies on the same plane.Examples of surfaces generated by a straight line are the cylinder and conical surfaces....
, Cartesian products with the real line, and a real affine transformation
Affine transformation

In geometry, an affine transformation or affine map or an affinity between two vector spaces consists of a linear transformation followed by a translation :...
.

Quadrics in Projective Geometry


In real projective space
Real projective space

In mathematics, real projective space, or RPn is the projective space of lines in Rn+1. It is a compact space, smooth manifold of dimension n, and a special case of a Grassmannian....
, the ellipsoid, the elliptic paraboloid and the hyperboloid of two sheets are equivalent to each other up to
Up to

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. "xxxx" describes a property or process which transforms an element into one from the same equivalence class, i.e....
 a projective transformation
Projective transformation

A projective transformation is a Transformation used in projective geometry: it is the composition of a pair of perspective projections. It describes what happens to the perceived positions of observed objects when the point of view of the observer changes....
; the hyperbolic paraboloid and the hyperboloid of one sheet are not different from each other (these are ruled surface
Ruled surface

In geometry, a surface is ruled if through every point of there is a straight line that lies on . The most familiar examples are the plane and the curved surface of a cylinder or cone ....
s); the cone and the cylinder are not different from each other (these are "degenerate" quadrics, since their Gaussian curvature
Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, ?1 and ?2, of the given point....
 is zero).

In complex projective space
Complex projective space

In mathematics, complex projective space, P, Pn or CPn, in fact preferablyis the projective space of line in Cn+1....
 all of the nondegenerate quadrics become indistinguishable from each other.

See also

  • Conic section
    Conic section

    File:Conic sections with plane.svgIn mathematics, a conic section is a curve obtained by intersecting a cone with a plane . A conic section is therefore a restriction of a quadric surface to the plane ....
  • Focus (geometry)
    Focus (geometry)

    In geometry, the foci, , are a pair of special points used in describing conic sections. The four types of conic sections are the circle, parabola, ellipse, and hyperbola....
    , an overview of properties of conic sections related to the foci.
  • Quadratic function
    Quadratic function

    A quadratic function, in mathematics, is a polynomial function of the form , where . The graph of a function of a quadratic function is a parabola whose major axis is parallel to the y-axis....


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