“Quadric”的意思、由来-开放百科全书

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 {{nowrap|(D + 1)}}-dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in D + 1 variables ({{nowrap|1=D = 1}} in the case of conic sections). When the defining polynomial is not absolutely irreducible, the zero set is generally not considered a quadric, although it is often called a degenerate quadric or a reducible quadric.

In coordinates {{nowrap|x₁, x₂, ..., x_D+1}}, the general quadric is thus defined by the algebraic equation^[1]

where {{nowrap|1=x = (x₁, x₂, ..., x_D+1)}} is a row vector, x^T is the transpose of x (a column vector), Q is a {{nowrap|(D + 1) × (D + 1)}} matrix and P is a {{nowrap|(D + 1)}}-dimensional row vector and R a scalar constant. The values Q, P and R are often taken to be over real numbers or complex numbers, but a quadric may be defined over any field.

A quadric is an affine algebraic variety, or, if it is reducible, an affine algebraic set. Quadrics may also be defined in projective spaces; see {{slink||Projective geometry}}, below.

Euclidean plane

Quadrics in the Euclidean plane are those of dimension D = 1, which is to say that they are plane curves. In this case, one talks of conic sections, or conics.

Euclidean space

In three-dimensional Euclidean space, quadrics have dimension D = 2, and are known as quadric surfaces. They are classified and named by their orbits under affine transformations. More precisely, if an affine transformation maps a quadric onto another one, they belong to the same class, and share the same name and many properties.

The principal axis theorem shows that for any (possibly reducible) quadric, a suitable Euclidean transformation or a change of Cartesian coordinates allows putting the quadratic equation of the quadric into one of the following normal forms:

corresponds to a single orbit under affine transformations. In three cases there are no real points:

(imaginary ellipsoid),

(imaginary elliptic cylinder), and

(pair of complex conjugate parallel planes, a reducible quadric). In one case, the imaginary cone, there is a single point (

). If

one has a line (in fact two complex conjugate intersecting planes). For

one has two intersecting planes (reducible quadric). For

one has a double plane. For

one has two parallel planes (reducible quadric).

Thus, among the 17 normal forms, there are nine true quadrics: a cone, three cylinders (often called degenerate quadrics) and five non-degenerate quadrics (ellipsoid, paraboloids and hyperboloids), which are detailed in the following tables. The eight remaining quadrics are the imaginary ellipsoid (no real point), the imaginary cylinder (no real point), the imaginary cone (a single real point), and the reducible quadrics, which are decomposed in two planes; there are five such decomposed quadrics, depending whether the planes are distinct or not, parallel or not, real or complex conjugate.

When two or more of the parameters of the canonical equation are equal, one gets a quadric of revolution, which remains invariant when rotated around an axis (or infinitely many axes, in the case of the sphere).

Definition and basic properties

A quadric is the set of zeros of a polynomial of degree two. When not specified otherwise, the polynomial is supposed to have real coefficients, and the zeros are points in a Euclidean space. However, most properties remain true when the coefficients belong to any field and the points belong in an affine space. As usually in algebraic geometry, it is often useful to consider points over an algebraically closed field containing the polynomial coefficients, generally the complex numbers, when the coefficients are real.

This define affine quadrics. Many properties becomes easier to state (and to prove) by extending the quadric to the projective space by projective completion consisting of adding points at infinity. Technically, if

is a polynomial of degree two that defines an affine quadric, then its projective completion is defined by homogenizing {{mvar|p}} into

(this is a polynomial, because the degree of {{mvar|p}} is two). The points of the projective completion are the points of the projective space whose projective coordinates are zeros of {{mvar|P}}.

Equation

A quadric in an affine space of dimension {{mvar|n}} is the set of zeros of a polynomial of degree 2, that is the set of the points whose coordinates satisfy an equation

where

if the characteristic of the field of the coefficients is not two and

otherwise.

If {{mvar|A}} is the {{math|(n + 1)×(n + 1)}} matrix that has the

as entries, and

These equations define a quartic as an algebraic hypersurface of dimension {{math|n – 1}} and degree two in a space of dimension {{mvar|n}}.

Normal form of projective quadrics

The quadrics can be treated in a uniform manner by introducing homogeneous coordinates on a Euclidean space, thus effectively regarding it as a projective space. Thus if the original (affine) coordinates on R^D+1 are

related to the original coordinates by

. In the new variables, every quadric is defined by an equation of the form

where the coefficients a_ij are symmetric in i and j. Regarding Q(X) = 0 as an equation in projective space exhibits the quadric as a projective algebraic variety. The quadric is said to be non-degenerate if the quadratic form is non-singular; equivalently, if the matrix (a_ij) is invertible.

In real projective space, by Sylvester's law of inertia, a non-singular quadratic form Q(X) may be put into the normal form

by means of a suitable projective transformation (normal forms for singular quadrics can have zeros as well as ±1 as coefficients). For surfaces in space (dimension D = 2) there are exactly three nondegenerate cases:

The second case generates the ellipsoid, the elliptic paraboloid or the hyperboloid of two sheets, depending on whether the chosen plane at infinity cuts the quadric in the empty set, in a point, or in a nondegenerate conic respectively. These all have positive Gaussian curvature.

The third case generates the hyperbolic paraboloid or the hyperboloid of one sheet, depending on whether the plane at infinity cuts it in two lines, or in a nondegenerate conic respectively. These are doubly ruled surfaces of negative Gaussian curvature.

generates the elliptic cylinder, the parabolic cylinder, the hyperbolic cylinder, or the cone, depending on whether the plane at infinity cuts it in a point, a line, two lines, or a nondegenerate conic respectively. These are singly ruled surfaces of zero Gaussian curvature.

We see that projective transformations don't mix Gaussian curvatures of different sign. This is true for general surfaces. ^[4]

In complex projective space all of the nondegenerate quadrics become indistinguishable from each other.

Projective quadrics over fields

The definition of a projective quadric in a real projective space (see above) can be formally adopted defining a projective quadric in an n-dimensional projective space over a field. In order to omit dealing with coordinates a projective quadric is usually defined starting with a quadratic form on a vector space ^[5]

Quadratic form

In case of

the bilinear form is

, i.e.

and

are mutually determined in a unique way.

n-dimensional projective space over a field

Projective quadric

For a quadratic form

on a vector space

a point

is called singular if

. The set

of singular points of

is called quadric (with respect to the quadratic form

(E2): For

one gets the pair of lines with the equations

and

, respectively. They intersect at point

;

Polar space

for at least one

, the equation

is a non trivial linear equation which defines a hyperplane. Hence

Intersection with a line

f-radical, q-radical

In the classical cases

there exists only one radical, because of

and

are closely connected. In case of

the quadric

is not determined by

(see above) and so one has to deal with two radicals:

In this case the f-radical is the common point of all tangents, the so called knot.

Symmetries

induces an involutorial central collineation

with axis

and centre

which leaves

invariant.

q-subspaces and index of a quadric

Generalization of quadrics: quadratic sets

It is not reasonable to extent formally the definition of quadrics to spaces over genuine skew fields (division rings). Because one would get secants bearing more than 2 points of the quadric which is totally different from usual quadrics ^[8]^[9]^[10]. The reason is the following statement.

There are generalizations of quadrics: quadratic sets^[11]. A quadratic set is a set of points of a projective space with the same geometric properties as a quadric: every line intersects a quadratic set in at most two points or is contained in the set.