# Quadric

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 (*D* + 1)-dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in *D* + 1 variables (*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 *x*_{1}, *x*_{2}, ..., *x*_{D+1}, the general quadric is thus defined by the algebraic equation[1]

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

where *x* = (*x*_{1}, *x*_{2}, ..., *x*_{D+1}) is a row vector, *x*^{T} is the transpose of *x* (a column vector), *Q* is a (*D* + 1) × (*D* + 1) matrix 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 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 § 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:

where the are either 1, –1 or 0, except which takes only the value 0 or 1.

Each of these 17 normal forms[2][3]
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.

Non-degenerate real quadric surfaces | ||
---|---|---|

Ellipsoid | ||

Elliptic paraboloid | ||

Hyperbolic paraboloid | ||

Elliptic hyperboloid of one sheet | ||

Elliptic hyperboloid of two sheets |

Degenerate real quadric surfaces | ||
---|---|---|

Elliptic cone | ||

Elliptic cylinder | ||

Hyperbolic cylinder | ||

Parabolic cylinder |

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).

Quadrics of revolution | ||
---|---|---|

Oblate and prolate spheroids (special cases of ellipsoid) | ||

Sphere (special case of spheroid) | ||

Circular paraboloid (special case of elliptic paraboloid) | ||

Circular hyperboloid of one sheet (special case of elliptic hyperboloid of one sheet) | ||

Circular hyperboloid of two sheets (special case of elliptic hyperboloid of two sheets) | ||

Circular cone (special case of elliptic cone) | ||

Circular cylinder (special case of elliptic cylinder) |

## 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 p into

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

### Equation

A quadric in an affine space of dimension 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 the polynomial p has the form

where if the characteristic of the field of the coefficients is not two and otherwise.

If A is the (*n* + 1)×(*n* + 1) matrix that has the as entries, and

then the equation may be shortened in the matrix equation

The equation of the projective completion of this quadric is

or

with

These equations define a quadric as an algebraic hypersurface of *dimension* *n* – 1 and degree two in a space of dimension 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

one introduces new coordinates on **R**^{D+2}

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 first case is the empty set.

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.

The degenerate form

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

Let be a field and a vector space over . A mapping from to such that

**(Q1)**for any and .

**(Q2)**is a bilinear form.

is called **quadratic form**. The bilinear form is symmetric*.*

In case of the bilinear form is , i.e. and are mutually determined in a unique way.

In case of (that means: ) the bilinear form has the property , i.e. is
*symplectic*.

For and ( is a base of ) has the familiar form

- and

- .

For example:

*n*-dimensional projective space over a field

*n*-dimensional projective space over a field

Let be a field, ,

- an (
*n*+ 1)-dimensional vector space over the field - the 1-dimensional subspace generated by ,
- the
*set of points*, - the
*set of lines*. - is the n-dimensional
**projective space**over . - The set of points contained in a -dimensional subspace of is a
*-dimensional subspace*of . A 2-dimensional subspace is a*plane*. - In case of a -dimensional subspace is called
*hyperplane*.

### 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 ).

**Examples in .:**

**(E1):** For one gets a conic.

**(E2):** For one gets the pair of lines with the equations and , respectively. They intersect at point ;

For the considerations below it is assumed that .

### Polar space

For point the set

is called **polar space** of (with respect to ).

If for any , one gets .

If for at least one , the equation is a non trivial linear equation which defines a hyperplane. Hence

- is either a hyperplane or .

### Intersection with a line

For the intersection of a line with a quadric the familiar statement is true:

- For an arbitrary line the following cases occur:
- a) and is called
*exterior line*or - b) and is called
*tangent line*or - b′) and is called
*tangent line*or - c) and is called
*secant line*.

**Proof:**
Let be a line, which intersects at point and is a second point on .
From one gets

I) In case of the equation holds and it is
for any . Hence either
for *any* or for *any* , which proves b) and b').

II) In case of one gets and the equation
has exactly one solution .
Hence: , which proves c).

Additionally the proof shows:

- A line through a point is a
*tangent*line if and only if .

*f*-radical, *q*-radical

*f*-radical,

*q*-radical

In the classical cases or there exists only one radical, because of and and are closely connected. In case of the quadric is not determined by (see above) and so one has to deal with two radicals:

- a) is a projective subspace. is called
of quadric .*f*-radical - b) is called
**singular radical**or*-radical*of . - c) In case of one has .

A quadric is called **non-degenerate** if .

**Examples in ** (see above):

**(E1):** For (conic) the bilinear form is

In case of the polar spaces are never . Hence .

In case of the bilinear form is reduced to
and . Hence
In this case the *f*-radical is the common point of all tangents, the so called *knot*.

In both cases and the quadric (conic) ist *non-degenerate*.

**(E2):** For (pair of lines) the bilinear form is and the intersection point.

In this example the quadric is *degenerate*.

### Symmetries

A quadric is a rather homogeneous object:

- For any point there exists an involutorial central collineation with center and .

**Proof:**
Due to the polar space is a hyperplane.

The linear mapping

induces an *involutorial central collineation* with axis and centre which leaves invariant.

In case of mapping gets the familiar shape with and for any .

**Remark:**

- a) An exterior line, a tangent line or a secant line is mapped by the involution on an exterior, tangent and secant line, respectively.
- b) is pointwise fixed by .

*q*-subspaces and index of a quadric

*q*-subspaces and index of a quadric

A subspace of is called -subspace if

For example: points on a sphere or lines on a hyperboloid (s. below).

- Any two
*maximal*-subspaces have the same dimension [6].

Let be the dimension of the maximal -subspaces of then

- The integer is called
**index**of .

**Theorem: (BUEKENHOUT)[7]**

- For the index of a non-degenerate quadric in the following is true:
- .

Let be a non-degenerate quadric in , and its index.

- In case of quadric is called
*sphere*(or oval conic if ). - In case of quadric is called
*hyperboloid*(of one sheet).

**Examples:**

- a) Quadric in with form is non-degenerate with index 1.
- b) If polynomial is irreducible over the quadratic form gives rise to a non-degenerate quadric in of index 1 (sphere). For example: is irreducible over (but not over !).
- c) In the quadratic form generates a
*hyperboloid*.

### Generalization of quadrics: quadratic sets

It is not reasonable to formally extend 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.

- A division ring is commutative if and only if any equation , has at most two solutions.

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.

## References

- Silvio Levy Quadrics in "Geometry Formulas and Facts", excerpted from 30th Edition of
*CRC Standard Mathematical Tables and Formulas*, CRC Press, from The Geometry Center at University of Minnesota - Stewart Venit and Wayne Bishop,
*Elementary Linear Algebra (fourth edition)*, International Thompson Publishing, 1996. - Sameen Ahmed Khan,Quadratic Surfaces in Science and Engineering, Bulletin of the IAPT, 2(11), 327–330 (November 2010). (Publication of the Indian Association of Physics Teachers).

Sameen Ahmed Khan, Coordinate Geometric Generalization of the Spherometer and Cylindrometer, arXiv:1311.3602

(Apparently, these unreliable sources forget one case) - S. Lazebnik and J. Ponce, "The Local Projective Shape of Smooth Surfaces and Their Outlines" (PDF)., Proposition 1
- Beutelspacher/Rosenbaum: p. 158
- Beutelpacher/Rosenbaum, p.139
- F. Buekenhout:
*Ensembles Quadratiques des Espace Projective*, Math. Teitschr. 110 (1969), p. 306-318. - R. Artzy:
*The Conic in Moufang Planes*, Aequat.Mathem. 6 (1971), p. 31-35 - E. Berz:
*Kegelschnitte in Desarguesschen Ebenen*, Math. Zeitschr. 78 (1962), p. 55-8 - external link E. Hartmann:
*Planar Circle Geometries*, p. 123 - Beutelspacher/Rosenbaum: p. 135

## Bibliography

- M. Audin:
*Geometry*, Springer, Berlin, 2002, ISBN 978-3-540-43498-6, p. 200. - M. Berger:
*Problem Books in Mathematics*, ISSN 0941-3502, Springer New York, pp 79-84. - A. Beutelspacher, U. Rosenbaum:
*Projektive Geometrie*, Vieweg + Teubner, Braunschweig u. a. 1992, ISBN 3-528-07241-5, p. 159. - P. Dembowski:
*Finite Geometries*, Springer, 1968, ISBN 978-3-540-61786-0, p. 43. - Iskovskikh, V.A. (2001) [1994], "Quadric", in Hazewinkel, Michiel (ed.),
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Weisstein, Eric W. "Quadric".
*MathWorld*.