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]}

- $\sum _{i,j=1}^{D+1}x_{i}Q_{ij}x_{j}+\sum _{i=1}^{D+1}P_{i}x_{i}+R=0$

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

- $xQx^{\mathrm {T} }+Px^{\mathrm {T} }+R=0\,$

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.

- 1 Euclidean plane
- 2 Euclidean space
- 3 Definition and basic properties
- 4 Normal form of projective quadrics
- 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:

- $xQx^{\mathrm {T} }+Px^{\mathrm {T} }+R=0\,$

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.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.

## Contents

- 1 Euclidean plane
- 2 Euclidean space
- 3 Definition and basic properties
- 3.1 Equation
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.

As the dimension of a Euclidean plane is two, quadrics in a Euclidean plane have dimension one and are thus plane curves. They are called

*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:

- $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}+{\epsilon}_{1}\frac{{z}^{2}}{}$
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:

- $\frac{{x}^{2}}{{a}^{}}$
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 $\varepsilon _{i}$ are either 1, –1 or 0, except $\varepsilon _{3}$ 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: $\varepsilon _{1}=\varepsilon _{2}=1$ (*imaginary ellipsoid*), $\varepsilon _{1}=0,\varepsilon _{2}=1$ (*imaginary elliptic cylinder*), and $\varepsilon _{4}=1$ (pair of complex conjugate parallel planes, a reducible quadric). In one case, the*imaginary cone*, there is a single point ($\varepsilon _{1}=1,\varepsilon _{2}=0$). If $\varepsilon _{1}=\varepsilon _{2}=0,$ one has a line (in fact two complex conjugate intersecting planes). For $\varepsilon _{3}=0,$ one has two intersecting planes (reducible quadric). For $\vareps$Each of these 17 normal forms

^{[2]}^{[3]}corresponds to a single orbit under affine transformations. In three cases there are no real points: $\varepsilon _{1}=\varepsilon _{2}=1$ (*imaginary ellipsoid*), $\varepsilon _{1}=0,\varepsilon _{2}=1$ (*imaginary elliptic cylinder*), and $\varepsilon _{4}=1$ (pair of complex conjugate parallel planes, a reducible quadric). In one case, the*imaginary cone*, there is a single point ($\varepsilon _{1}=1,\varepsilon _{2}=0$). If $\varepsilon _{1}=\varepsilon _{2}=0,$ one has a line (in fact two complex conjugate intersecting planes). For $\varepsilon _{3}=0,$ one has two intersecting planes (reducible quadric). For $\varepsilon _{4}=0,$ one has a double plane. For $\varepsilon _{4}=-1,$ 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

An

*affine 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.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

- $p(x_{1},\ldots ,x_{n})$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.
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

- $p(x_{1},\ldots ,x_{n})$

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

- $P(X_{0},\ldots ,X_{n})=X_{0}^{2}\,p\left({\frac {X_{1}}{X_{0}}},\ldots ,{\frac {X_{n}}{X_{0}}}\right)$

(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.

So, a

*projective quadric*is the set of zeros in a projective space of a homogeneous polynomial of degree two.As the above process of homogenization can be reverted by setting

*X*_{0}= 1, it is often useful to not distinguish an affine quadric from its projective completion, and to talk of the*affine equation*or the*projective equation*of a quadric.### 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

- $p(x_{1},\ldots ,x_{n})=0,$

where the polynomial p has the form

- $p(x_{1},\ldots ,x_{n})=\sum _{i=1}^{n}a_{i,i}x_{i}^{2}+\sum _{1\leq i<j\leq n}(a_{i,j}+a_{j,i})x_{i}x_{j}+\sum _{i=1}^{n}(a_{i,0}+a_{0,i})x_{i}+a_{0,0},$

where $a_{i,j}=a_{j,i}$ if the characteristic of the field of the coefficients is not two and $a_{j,i}=0$ otherwise.

If A is the (

*n*+ 1)×(*n*+ 1) matrix that has the $a_{i,j}$ as entries, and- $\mathbf {x} ={\begin{pmatrix}1&x_{1}&\cdots &x_{n}\end{pmatrix}}^{\mathsf {T}},$

then the equation may be shortened in the matrix equation

- $\mathbf {x} ^{\mathsf {T}}A\mathbf {x} =0.$

The equation of the projective completion of this quadric is

- $$
is a polynomial of degree two that defines an affine quadric, then its projective completion is defined by homogenizing p into

- $P({X}_{0},\dots ,{X}_{n})={X}_{0}^{2}\phantom{\rule{thinmathspace}{0ex}}p((this\; is\; a\; polynomial,\; because\; the\; degree\; ofp$ is two). The points of the projective completion are the points of the projective space whose projective coordinates are zeros of P.
So, a

*projective quadric*is the set of zeros in a projective space of a homogeneous polynomial of degree two.As the above process of homogenization can be reverted by setting

*X*_{0}= 1, it is often useful to not distinguish an affine quadric from its projective completion, and to talk of the*affine equation*or the*projective equation*of a quadric.### 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

- $p(x_{1},\ldots ,x_{n})=0,$

where the polynomial p has the form

- $$
As the above process of homogenization can be reverted by setting

*X*_{0}= 1, it is often useful to not distinguish an affine quadric from its projective completion, and to talk of the*affine equation*or the*projective equation*of a quadric.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
- $a_{i,j}=a_{j,i}$

- $where\; the\; polynomial$p has the form

- $P({X}_{0},\dots ,{X}_{n})={X}_{0}^{2}\phantom{\rule{thinmathspace}{0ex}}p((this\; is\; a\; polynomial,\; because\; the\; degree\; ofp$ is two). The points of the projective completion are the points of the projective space whose projective coordinates are zeros of P.

- $p(x_{1},\ldots ,x_{n})$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.

- $\frac{{x}^{2}}{{a}^{}}$

- $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}+{\epsilon}_{1}\frac{{z}^{2}}{}$

- 3.1 Equation