Quadric
Encyclopedia
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...

 in (D + 1)-dimensional space defined as the locus
Locus (mathematics)
In geometry, a locus is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point....

 of zeros of a quadratic polynomial
Quadratic polynomial
In mathematics, a quadratic polynomial or quadratic is a polynomial of degree two, also called second-order polynomial. That means the exponents of the polynomial's variables are no larger than 2...

. In coordinates }, the general quadric is defined by the algebraic equation


which may be compactly written in vector and matrix notation as:


where x = } is a row vector, xT is the transpose
Transpose
In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...

 of x (a column vector), Q is a (D + 1)×(D + 1) matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

 and P is a (D + 1)-dimensional row vector and R a scalar constant. The values Q, P and R are often taken to be real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s or complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s, but in fact, a quadric may be defined over any ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

. In general, the locus of zeros of a set of polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

s is known as an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...

, and is studied in the branch of algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

.

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...

.

Euclidean plane and space

Quadrics in the Euclidean plane are those of dimension D = 1, which is to say that they are curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

s. Such quadrics are the same as conic section
Conic section
In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...

s, and are typically known as conics rather than quadrics.
In Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

, quadrics have dimension D = 2, and are known as quadric surfaces. By making a suitable Euclidean change of variables, any quadric in Euclidean space can be put into a certain normal form by choosing as the coordinate directions the principal axes
Principal axis theorem
In the mathematical fields of geometry and linear algebra, a principal axis is a certain line in a Euclidean space associated to an ellipsoid or hyperboloid, generalizing the major and minor axes of an ellipse...

 of the quadric. In three-dimensional Euclidean space there are 16 such normal forms. Of these 16 forms, five are nondegenerate, and the remaining are degenerate forms. Degenerate forms include plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

s, line
Line (mathematics)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...

s, points or even no points at all.
Non-degenerate real quadric surfaces
Ellipsoid
    Spheroid
Spheroid
A spheroid, or ellipsoid of revolution is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters....

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

 (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 of one sheet
Hyperboloid of two sheets
Degenerate quadric surfaces
Cone
Conical surface
In geometry, a conical surface is the unbounded surface formed by the union of all the straight lines 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...

    Circular Cone
Conical surface
In geometry, a conical surface is the unbounded surface formed by the union of all the straight lines 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...

 (special case of cone)
Elliptic cylinder
Cylinder (geometry)
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...

    Circular cylinder
Cylinder (geometry)
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...

 (special case of elliptic cylinder)
Hyperbolic cylinder
Cylinder (geometry)
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...

Parabolic cylinder
Cylinder (geometry)
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...


Projective geometry

The quadrics can be treated in a uniform manner by introducing homogeneous coordinates
Homogeneous coordinates
In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...

 on a Euclidean space, thus effectively regarding it as a projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....

. Thus if the original (affine) coordinates on RD+1 are
one introduces new coordinates on RD+2
related to the original coordinates by . In the new variables, every quadric is defined by an equation of the form
where the coefficients aij are symmetric in i and j. Regarding Q(X) = 0 as an equation in projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....

 exhibits the quadric as a projective algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...

. The quadric is said to be non-degenerate if the quadratic form is non-singular; equivalently, if the matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

 (aij) is invertible.

In real projective space
Real projective space
In mathematics, real projective space, or RPn, is the topological space of lines through 0 in Rn+1. It is a compact, smooth manifold of dimension n, and a special case of a Grassmannian.-Construction:...

, by Sylvester's law of inertia
Sylvester's law of inertia
Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of coordinates...

, a non-singular quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....

 Q(X) may be put into the normal form


by means of a suitable projective transformation. The normal forms of singular quadrics can have zeros as well as ±1 for coefficients. This normal form thus classifies real non-singular quadrics up to
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in  x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...

 a projective transformation. In dimension D = 2, there are exactly three inequivalent cases:


Consequently, the ellipsoid, the elliptic paraboloid and the hyperboloid of two sheets are equivalent to each other up to a projective transformation as these all correspond to the second normal form. The hyperbolic paraboloid and the hyperboloid of one sheet are both equivalent to the third form (these are ruled surface
Ruled surface
In geometry, a surface S is ruled if through every point of S there is a straight line that lies on S. The most familiar examples are the plane and the curved surface of a cylinder or cone...

s). The cone and the cylinder are equivalent to the degenerate form.
These latter are degenerate quadrics because the coefficient on X3 is zero; they also have zero 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, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way...

.

In complex projective space
Complex projective space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines...

 all of the nondegenerate quadrics become indistinguishable from each other.

See also

  • Superquadrics
    Superquadrics
    In mathematics, the superquadrics or super-quadrics are a family of geometric shapes defined by formulas that resemble those of elipsoids and other quadrics, except that the squaring operations are replaced by arbitrary powers...

  • Conic section
    Conic section
    In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...

  • Focus (geometry)
    Focus (geometry)
    In geometry, the foci are a pair of special points with reference to which any of a variety of curves is constructed. For example, foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, 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 formf=ax^2+bx+c,\quad a \ne 0.The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y-axis....

  • Klein quadric
    Klein quadric
    The lines of a 3-dimensional projective space, S, can be viewed as points of a 5-dimensional projective space, T. In that 5-space, the points that represent a line in S lie on a hyperbolic quadric, Q known as the Klein quadric....


External links

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