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 x₁, x₂, ..., x_D+1, the general quadric is thus defined by the algebraic equation^[1]

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

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

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

where x = (x₁, x₂, ..., 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[]

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 two, and are known as quadric surfaces. Their quadratic equations have the form

${\displaystyle Ax^{2}+By^{2}+Cz^{2}+Dxy+Eyz+Fxz+Gx+Hy+Iz+J=0,}$

where ${\displaystyle A,B,\ldots ,J}$ are real numbers, and at least one of A, B, and C is nonzero.

The quadric surfaces are classified and named by their shape, which corresponds to the orbits under affine transformations. That is, 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 change of Cartesian coordinates or, equivalently, a Euclidean transformation allows putting the equation of the quadric into a unique simple form on which the class of the quadric is immediately visible. This form is called the normal form of the equation, since two quadrics have the same normal form if and only if there is a Euclidean transformation that maps one quadric to the other. The normal forms are as follows:

${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}+\varepsilon _{1}{z^{2} \over c^{2}}+\varepsilon _{2}=0,}$

${\displaystyle {x^{2} \over a^{2}}-{y^{2} \over b^{2}}+\varepsilon _{3}=0}$

${\displaystyle {x^{2} \over a^{2}}+\varepsilon _{4}=0,}$

${\displaystyle z={x^{2} \over a^{2}}+\varepsilon _{5}{y^{2} \over b^{2}},}$

where the ${\displaystyle \varepsilon _{i}}$ are either 1, –1 or 0, except ${\displaystyle \varepsilon _{3}}$ which takes only the value 0 or 1.

Each of these 17 normal forms^[2] corresponds to a single orbit under affine transformations. In three cases there are no real points: ${\displaystyle \varepsilon _{1}=\varepsilon _{2}=1}$ (imaginary ellipsoid), ${\displaystyle \varepsilon _{1}=0,\varepsilon _{2}=1}$ (imaginary elliptic cylinder), and ${\displaystyle \varepsilon _{4}=1}$ (pair of complex conjugate parallel planes, a reducible quadric). In one case, the imaginary cone, there is a single point (${\displaystyle \varepsilon _{1}=1,\varepsilon _{2}=0}$). If ${\displaystyle \varepsilon _{1}=\varepsilon _{2}=0,}$ one has a line (in fact two complex conjugate intersecting planes). For ${\displaystyle \varepsilon _{3}=0,}$ one has two intersecting planes (reducible quadric). For ${\displaystyle \varepsilon _{4}=0,}$ one has a double plane. For ${\displaystyle \varepsilon _{4}=-1,}$ 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.

Non-degenerate real quadric surfaces
Ellipsoid	${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}+{z^{2} \over c^{2}}=1\,}$
Elliptic paraboloid	${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}-z=0\,}$
Hyperbolic paraboloid	${\displaystyle {x^{2} \over a^{2}}-{y^{2} \over b^{2}}-z=0\,}$
Hyperboloid of one sheet       or    Hyperbolic hyperboloid	${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=1\,}$
Hyperboloid of two sheets       or    Elliptic hyperboloid	${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=-1\,}$

Degenerate real quadric surfaces
Elliptic cone       or    Conical quadric	${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=0\,}$
Elliptic cylinder	${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}=1\,}$
Hyperbolic cylinder	${\displaystyle {x^{2} \over a^{2}}-{y^{2} \over b^{2}}=1\,}$
Parabolic cylinder	${\displaystyle x^{2}+2ay=0\,}$

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)	${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over a^{2}}+{z^{2} \over b^{2}}=1\,}$
Sphere (special case of spheroid)	${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over a^{2}}+{z^{2} \over a^{2}}=1\,}$
Circular paraboloid (special case of elliptic paraboloid)	${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over a^{2}}-z=0\,}$
Hyperboloid of revolution of one sheet (special case of hyperboloid of one sheet)	${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over a^{2}}-{z^{2} \over b^{2}}=1\,}$
Hyperboloid of revolution of two sheets (special case of hyperboloid of two sheets)	${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over a^{2}}-{z^{2} \over b^{2}}=-1\,}$
Circular cone (special case of elliptic cone)	${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over a^{2}}-{z^{2} \over b^{2}}=0\,}$
Circular cylinder (special case of elliptic cylinder)	${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over a^{2}}=1\,}$

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

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

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

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

(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 X₀ = 1:

${\displaystyle p(x_{1},\ldots ,x_{n})=P(1,x_{1},\ldots ,x_{n})\,,}$

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. However, this is not a perfect equivalence; it is generally the case that ${\displaystyle P(\mathbf {X} )=0}$ will include points with ${\displaystyle X_{0}=0}$, which are not also solutions of ${\displaystyle p(\mathbf {x} )=0}$ because these points in projective space correspond to points "at infinity" in affine space.

Equation[]

A quadric in an affine space of dimension n is the set of zeros of a polynomial of degree 2. That is, it is the set of the points whose coordinates satisfy an equation

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

where the polynomial p has the form

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

for a matrix ${\displaystyle A=(a_{i,j})}$ with ${\displaystyle i}$ and ${\displaystyle j}$ running from 0 to ${\displaystyle n}$. When the characteristic of the field of the coefficients is not two, generally ${\displaystyle a_{i,j}=a_{j,i}}$ is assumed; equivalently ${\displaystyle A=A^{\mathsf {T}}}$. When the characteristic of the field of the coefficients is two, generally ${\displaystyle a_{i,j}=0}$ is assumed when ${\displaystyle j<i}$; equivalently ${\displaystyle A}$ is upper triangular.

The equation may be shortened, as the matrix equation

${\displaystyle \mathbf {x} ^{\mathsf {T}}A\mathbf {x} =0\,,}$

with

${\displaystyle \mathbf {x} ={\begin{pmatrix}1&x_{1}&\cdots &x_{n}\end{pmatrix}}^{\mathsf {T}}\,.}$

The equation of the projective completion is almost identical:

${\displaystyle \mathbf {X} ^{\mathsf {T}}A\mathbf {X} =0,}$

with

${\displaystyle \mathbf {X} ={\begin{pmatrix}X_{0}&X_{1}&\cdots &X_{n}\end{pmatrix}}^{\mathsf {T}}.}$

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

The quadric is said to be non-degenerate if the matrix ${\displaystyle A}$ is invertible.

Normal form of projective quadrics[]

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

${\displaystyle P(X)=\pm X_{0}^{2}\pm X_{1}^{2}\pm \cdots \pm X_{D+1}^{2}}$

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

${\displaystyle P(X)={\begin{cases}X_{0}^{2}+X_{1}^{2}+X_{2}^{2}+X_{3}^{2}\\X_{0}^{2}+X_{1}^{2}+X_{2}^{2}-X_{3}^{2}\\X_{0}^{2}+X_{1}^{2}-X_{2}^{2}-X_{3}^{2}\end{cases}}}$

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

${\displaystyle X_{0}^{2}-X_{1}^{2}-X_{2}^{2}=0.\,}$

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.^[3]

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

Integer and rational solutions[]

Each solution of ${\displaystyle p(\mathbf {x} )=0}$ with ${\displaystyle \mathbf {x} }$ a vector having rational components yields a vector ${\displaystyle \mathbf {X} \neq \mathbf {0} }$ with integer components that satisfies ${\displaystyle P(\mathbf {X} )=0}$; set ${\displaystyle \mathbf {X} =k(1,\mathbf {x} )}$ where the multiplying factor ${\displaystyle k}$ is the smallest positive integer that clears all the denominators of the components of ${\displaystyle \mathbf {x} }$.

Furthermore, when the underlying matrix is invertible, any one solution to ${\displaystyle p(\mathbf {y} )=0}$ for ${\displaystyle \mathbf {y} }$ with rational components can be used to find any other solution with rational components, as follows. Let ${\displaystyle \mathbf {X} =\mathbf {Y} +\lambda \mathbf {Z} }$ for some values of ${\displaystyle \mathbf {Y} \neq \mathbf {0} }$ and ${\displaystyle \mathbf {Z} \neq \mathbf {0} }$, both with integer components, and value ${\displaystyle \lambda }$. Writing ${\displaystyle P(\mathbf {X} )=\mathbf {X} ^{\mathsf {T}}A\mathbf {X} }$ for a non-singular symmetric matrix ${\displaystyle A}$ with integer components, we have that

${\displaystyle P(\mathbf {X} )=\mathbf {Y} ^{\mathsf {T}}A\mathbf {Y} +2\lambda \mathbf {Y} ^{\mathsf {T}}A\mathbf {Z} +\lambda ^{2}\mathbf {Z} ^{\mathsf {T}}A\mathbf {Z} \,.}$

When

${\displaystyle {\begin{aligned}\mathbf {Y} ^{\mathsf {T}}A\mathbf {Y} &=0,\\\mathbf {Y} ^{\mathsf {T}}A\mathbf {Z} &\neq 0,\\\mathbf {Z} ^{\mathsf {T}}A\mathbf {Z} &\neq 0\end{aligned}}}$

then the two solutions to ${\displaystyle P(\mathbf {X} )=0}$, when viewed as a quadratic equation in ${\displaystyle \lambda }$, will be ${\displaystyle \lambda =0,-(2\mathbf {Y} ^{\mathsf {T}}A\mathbf {Z} )/(\mathbf {Z} ^{\mathsf {T}}A\mathbf {Z} )}$, where the latter is non-zero and rational. In particular, if ${\displaystyle \mathbf {y} }$ is a solution of ${\displaystyle p(\mathbf {y} )=0}$ and ${\displaystyle \mathbf {Y} }$ is the corresponding non-zero solution of ${\displaystyle P(\mathbf {Y} )=0}$ then any ${\displaystyle \mathbf {Z} }$ for which (1) ${\displaystyle \mathbf {Z} }$ is not orthogonal to ${\displaystyle A\mathbf {Y} }$ and (2) ${\displaystyle \mathbf {Z} ^{\mathsf {T}}A\mathbf {Z} \neq 0}$ satisfies these three conditions and gives a non-zero rational value for ${\displaystyle \lambda }$.

In short, if one knows one solution ${\displaystyle \mathbf {y} }$ with rational components then one can find many integer solutions ${\displaystyle \mathbf {W} _{\mathbf {Z} }=(\mathbf {Z} ^{\mathsf {T}}A\mathbf {Z} )\mathbf {Y} -(2\mathbf {Y} ^{\mathsf {T}}A\mathbf {Z} )\mathbf {Z} }$ where ${\displaystyle \mathbf {W} _{\mathbf {Z} }}$ depends upon the choice of ${\displaystyle \mathbf {Z} }$. Furthermore, the process is reversible! If both ${\displaystyle \mathbf {y} }$ satisfies ${\displaystyle p(\mathbf {y} )=0}$ and ${\displaystyle \mathbf {w} }$ satisfies ${\displaystyle p(\mathbf {w} )=0}$ then the choice of ${\displaystyle \mathbf {Z} =\mathbf {W} -\mathbf {Y} }$ will necessarily produce ${\displaystyle \mathbf {W} _{\mathbf {Z} }=\mathbf {W} }$. With this approach one can generate all Pythagorean triples or Heronian triangles.

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 ^[4]

Quadratic form[]

Let ${\displaystyle K}$ be a field and ${\displaystyle V}$ a vector space over ${\displaystyle K}$. A mapping ${\displaystyle q}$ from ${\displaystyle V}$ to ${\displaystyle K}$ such that

(Q1) ${\displaystyle \;q(\lambda {\vec {x}})=\lambda ^{2}q({\vec {x}})\;}$ for any ${\displaystyle \lambda \in K}$ and ${\displaystyle {\vec {x}}\in V}$.

(Q2) ${\displaystyle \;f({\vec {x}},{\vec {y}}):=q({\vec {x}}+{\vec {y}})-q({\vec {x}})-q({\vec {y}})\;}$ is a bilinear form.

is called quadratic form. The bilinear form ${\displaystyle f}$ is symmetric.

In case of ${\displaystyle \operatorname {char} K\neq 2}$ the bilinear form is ${\displaystyle f({\vec {x}},{\vec {x}})=2q({\vec {x}})}$, i.e. ${\displaystyle f}$ and ${\displaystyle q}$ are mutually determined in a unique way.
In case of ${\displaystyle \operatorname {char} K=2}$ (that means: ${\displaystyle 1+1=0}$) the bilinear form has the property ${\displaystyle f({\vec {x}},{\vec {x}})=0}$, i.e. ${\displaystyle f}$ is symplectic.

For ${\displaystyle V=K^{n}\ }$ and ${\displaystyle \ {\vec {x}}=\sum _{i=1}^{n}x_{i}{\vec {e}}_{i}\quad }$ (${\displaystyle \{{\vec {e}}_{1},\ldots ,{\vec {e}}_{n}\}}$ is a base of ${\displaystyle V}$) ${\displaystyle \ q}$ has the familiar form

${\displaystyle q({\vec {x}})=\sum _{1=i\leq k}^{n}a_{ik}x_{i}x_{k}\ {\text{ with }}\ a_{ik}:=f({\vec {e}}_{i},{\vec {e}}_{k})\ {\text{ for }}\ i\neq k\ {\text{ and }}\ a_{ii}:=q({\vec {e}}_{i})\ }$ and

${\displaystyle f({\vec {x}},{\vec {y}})=\sum _{1=i\leq k}^{n}a_{ik}(x_{i}y_{k}+x_{k}y_{i})}$.

For example:

${\displaystyle n=3,\quad q({\vec {x}})=x_{1}x_{2}-x_{3}^{2},\quad f({\vec {x}},{\vec {y}})=x_{1}y_{2}+x_{2}y_{1}-2x_{3}y_{3}\;.}$

n-dimensional projective space over a field[]

Let ${\displaystyle K}$ be a field, ${\displaystyle 2\leq n\in \mathbb {N} }$,

${\displaystyle V_{n+1}}$ an (n + 1)-dimensional vector space over the field ${\displaystyle K,}$

${\displaystyle \langle {\vec {x}}\rangle }$ the 1-dimensional subspace generated by ${\displaystyle {\vec {0}}\neq {\vec {x}}\in V_{n+1}}$,

${\displaystyle {\mathcal {P}}=\{\langle {\vec {x}}\rangle \mid {\vec {x}}\in V_{n+1}\},\ }$ the set of points ,

${\displaystyle {\mathcal {G}}=\{{\text{2-dimensional subspaces of }}V_{n+1}\},\ }$ the set of lines.

${\displaystyle P_{n}(K)=({\mathcal {P}},{\mathcal {G}})\ }$ is the n-dimensional projective space over ${\displaystyle K}$.

The set of points contained in a ${\displaystyle (k+1)}$-dimensional subspace of ${\displaystyle V_{n+1}}$ is a ${\displaystyle k}$-dimensional subspace of ${\displaystyle P_{n}(K)}$. A 2-dimensional subspace is a plane.

In case of ${\displaystyle \;n>3\;}$ a ${\displaystyle (n-1)}$-dimensional subspace is called hyperplane.

Projective quadric[]

For a quadratic form ${\displaystyle q}$ on a vector space ${\displaystyle V_{n+1}}$ a point ${\displaystyle \langle {\vec {x}}\rangle \in {\mathcal {P}}}$ is called singular if ${\displaystyle q({\vec {x}})=0}$. The set

${\displaystyle {\mathcal {Q}}=\{\langle {\vec {x}}\rangle \in {\mathcal {P}}\mid q({\vec {x}})=0\}}$

of singular points of ${\displaystyle q}$ is called quadric (with respect to the quadratic form ${\displaystyle q}$).

Examples in ${\displaystyle P_{2}(K)}$.:
(E1): For ${\displaystyle \;q({\vec {x}})=x_{1}x_{2}-x_{3}^{2}\;}$ one gets a conic.
(E2): For ${\displaystyle \;q({\vec {x}})=x_{1}x_{2}\;}$ one gets the pair of lines with the equations ${\displaystyle x_{1}=0}$ and ${\displaystyle x_{2}=0}$, respectively. They intersect at point ${\displaystyle \langle (0,0,1)^{\text{T}}\rangle }$;

For the considerations below it is assumed that ${\displaystyle {\mathcal {Q}}\neq \emptyset }$.

Polar space[]

For point ${\displaystyle P=\langle {\vec {p}}\rangle \in {\mathcal {P}}}$ the set

${\displaystyle P^{\perp }:=\{\langle {\vec {x}}\rangle \in {\mathcal {P}}\mid f({\vec {p}},{\vec {x}})=0\}}$

is called polar space of ${\displaystyle P}$ (with respect to ${\displaystyle q}$).

If ${\displaystyle \;f({\vec {p}},{\vec {x}})=0\;}$ for any ${\displaystyle {\vec {x}}}$, one gets ${\displaystyle P^{\perp }={\mathcal {P}}}$.

If ${\displaystyle \;f({\vec {p}},{\vec {x}})\neq 0\;}$ for at least one ${\displaystyle {\vec {x}}}$, the equation ${\displaystyle \;f({\vec {p}},{\vec {x}})=0\;}$is a non trivial linear equation which defines a hyperplane. Hence

${\displaystyle P^{\perp }}$ is either a hyperplane or ${\displaystyle {\mathcal {P}}}$.

Intersection with a line[]

For the intersection of a line with a quadric ${\displaystyle {\mathcal {Q}}}$ the familiar statement is true:

For an arbitrary line ${\displaystyle g}$ the following cases occur:

a) ${\displaystyle g\cap {\mathcal {Q}}=\emptyset \;}$ and ${\displaystyle g}$ is called exterior line or

b) ${\displaystyle g\subset {\mathcal {Q}}\;}$ and ${\displaystyle g}$ is called tangent line or

b′) ${\displaystyle |g\cap {\mathcal {Q}}|=1\;}$ and ${\displaystyle g}$ is called tangent line or

c) ${\displaystyle |g\cap {\mathcal {Q}}|=2\;}$ and ${\displaystyle g}$ is called secant line.

Proof: Let ${\displaystyle g}$ be a line, which intersects ${\displaystyle {\mathcal {Q}}}$ at point ${\displaystyle \;U=\langle {\vec {u}}\rangle \;}$ and ${\displaystyle \;V=\langle {\vec {v}}\rangle \;}$ is a second point on ${\displaystyle g}$. From ${\displaystyle \;q({\vec {u}})=0\;}$ one gets
${\displaystyle q(x{\vec {u}}+{\vec {v}})=q(x{\vec {u}})+q({\vec {v}})+f(x{\vec {u}},{\vec {v}})=q({\vec {v}})+xf({\vec {u}},{\vec {v}})\;.}$
I) In case of ${\displaystyle g\subset U^{\perp }}$ the equation ${\displaystyle f({\vec {u}},{\vec {v}})=0}$ holds and it is ${\displaystyle \;q(x{\vec {u}}+{\vec {v}})=q({\vec {v}})\;}$ for any ${\displaystyle x\in K}$. Hence either ${\displaystyle \;q(x{\vec {u}}+{\vec {v}})=0\;}$ for any ${\displaystyle x\in K}$ or ${\displaystyle \;q(x{\vec {u}}+{\vec {v}})\neq 0\;}$ for any ${\displaystyle x\in K}$, which proves b) and b').
II) In case of ${\displaystyle g\not \subset U^{\perp }}$ one gets ${\displaystyle f({\vec {u}},{\vec {v}})\neq 0}$ and the equation ${\displaystyle \;q(x{\vec {u}}+{\vec {v}})=q({\vec {v}})+xf({\vec {u}},{\vec {v}})=0\;}$ has exactly one solution ${\displaystyle x}$. Hence: ${\displaystyle |g\cap {\mathcal {Q}}|=2}$, which proves c).

Additionally the proof shows:

A line ${\displaystyle g}$ through a point ${\displaystyle P\in {\mathcal {Q}}}$ is a tangent line if and only if ${\displaystyle g\subset P^{\perp }}$.

f-radical, q-radical[]

In the classical cases ${\displaystyle K=\mathbb {R} }$ or ${\displaystyle \mathbb {C} }$ there exists only one radical, because of ${\displaystyle \operatorname {char} K\neq 2}$ and ${\displaystyle f}$ and ${\displaystyle q}$ are closely connected. In case of ${\displaystyle \operatorname {char} K=2}$ the quadric ${\displaystyle {\mathcal {Q}}}$ is not determined by ${\displaystyle f}$ (see above) and so one has to deal with two radicals:

a) ${\displaystyle {\mathcal {R}}:=\{P\in {\mathcal {P}}\mid P^{\perp }={\mathcal {P}}\}}$ is a projective subspace. ${\displaystyle {\mathcal {R}}}$ is called f-radical of quadric ${\displaystyle {\mathcal {Q}}}$.

b) ${\displaystyle {\mathcal {S}}:={\mathcal {R}}\cap {\mathcal {Q}}}$ is called singular radical or ${\displaystyle q}$-radical of ${\displaystyle {\mathcal {Q}}}$.

c) In case of ${\displaystyle \operatorname {char} K\neq 2}$ one has ${\displaystyle {\mathcal {R}}={\mathcal {S}}}$.

A quadric is called non-degenerate if ${\displaystyle {\mathcal {S}}=\emptyset }$.

Examples in ${\displaystyle P_{2}(K)}$ (see above):
(E1): For ${\displaystyle \;q({\vec {x}})=x_{1}x_{2}-x_{3}^{2}\;}$ (conic) the bilinear form is ${\displaystyle f({\vec {x}},{\vec {y}})=x_{1}y_{2}+x_{2}y_{1}-2x_{3}y_{3}\;.}$
In case of ${\displaystyle \operatorname {char} K\neq 2}$ the polar spaces are never ${\displaystyle {\mathcal {P}}}$. Hence ${\displaystyle {\mathcal {R}}={\mathcal {S}}=\emptyset }$.
In case of ${\displaystyle \operatorname {char} K=2}$ the bilinear form is reduced to ${\displaystyle f({\vec {x}},{\vec {y}})=x_{1}y_{2}+x_{2}y_{1}\;}$ and ${\displaystyle {\mathcal {R}}=\langle (0,0,1)^{\text{T}}\rangle \notin {\mathcal {Q}}}$. Hence ${\displaystyle {\mathcal {R}}\neq {\mathcal {S}}=\emptyset \;.}$ In this case the f-radical is the common point of all tangents, the so called knot.
In both cases ${\displaystyle S=\emptyset }$ and the quadric (conic) ist non-degenerate.
(E2): For ${\displaystyle \;q({\vec {x}})=x_{1}x_{2}\;}$ (pair of lines) the bilinear form is ${\displaystyle f({\vec {x}},{\vec {y}})=x_{1}y_{2}+x_{2}y_{1}\;}$ and ${\displaystyle {\mathcal {R}}=\langle (0,0,1)^{\text{T}}\rangle ={\mathcal {S}}\;,}$ the intersection point.
In this example the quadric is degenerate.

Symmetries[]

A quadric is a rather homogeneous object:

For any point ${\displaystyle P\notin {\mathcal {Q}}\cup {\mathcal {R}}\;}$ there exists an involutorial central collineation ${\displaystyle \sigma _{P}}$ with center ${\displaystyle P}$ and ${\displaystyle \sigma _{P}({\mathcal {Q}})={\mathcal {Q}}}$.

Proof: Due to ${\displaystyle P\notin {\mathcal {Q}}\cup {\mathcal {R}}}$ the polar space ${\displaystyle P^{\perp }}$ is a hyperplane.

The linear mapping

${\displaystyle \varphi :{\vec {x}}\rightarrow {\vec {x}}-{\frac {f({\vec {p}},{\vec {x}})}{q({\vec {p}})}}{\vec {p}}}$

induces an involutorial central collineation ${\displaystyle \sigma _{P}}$ with axis ${\displaystyle P^{\perp }}$ and centre ${\displaystyle P}$ which leaves ${\displaystyle {\mathcal {Q}}}$ invariant.
In case of ${\displaystyle \operatorname {char} K\neq 2}$ mapping ${\displaystyle \varphi }$ gets the familiar shape ${\displaystyle \;\varphi :{\vec {x}}\rightarrow {\vec {x}}-2{\frac {f({\vec {p}},{\vec {x}})}{f({\vec {p}},{\vec {p}})}}{\vec {p}}\;}$ with ${\displaystyle \;\varphi ({\vec {p}})=-{\vec {p}}}$ and ${\displaystyle \;\varphi ({\vec {x}})={\vec {x}}\;}$ for any ${\displaystyle \langle {\vec {x}}\rangle \in P^{\perp }}$.

Remark:

a) An exterior line, a tangent line or a secant line is mapped by the involution ${\displaystyle \sigma _{P}}$ on an exterior, tangent and secant line, respectively.

b) ${\displaystyle {\mathcal {R}}}$ is pointwise fixed by ${\displaystyle \sigma _{P}}$.

q-subspaces and index of a quadric[]

A subspace ${\displaystyle \;{\mathcal {U}}\;}$ of ${\displaystyle P_{n}(K)}$ is called ${\displaystyle q}$-subspace if ${\displaystyle \;{\mathcal {U}}\subset {\mathcal {Q}}\;}$

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

Any two maximal ${\displaystyle q}$-subspaces have the same dimension ${\displaystyle m}$.^[5]

Let be ${\displaystyle m}$ the dimension of the maximal ${\displaystyle q}$-subspaces of ${\displaystyle {\mathcal {Q}}}$ then

The integer ${\displaystyle \;i:=m+1\;}$ is called index of ${\displaystyle {\mathcal {Q}}}$.

Theorem: (BUEKENHOUT)^[6]

For the index ${\displaystyle i}$ of a non-degenerate quadric ${\displaystyle {\mathcal {Q}}}$ in ${\displaystyle P_{n}(K)}$ the following is true:

${\displaystyle i\leq {\frac {n+1}{2}}}$.

Let be ${\displaystyle {\mathcal {Q}}}$ a non-degenerate quadric in ${\displaystyle P_{n}(K),n\geq 2}$, and ${\displaystyle i}$ its index.

In case of ${\displaystyle i=1}$ quadric ${\displaystyle {\mathcal {Q}}}$ is called sphere (or oval conic if ${\displaystyle n=2}$).

In case of ${\displaystyle i=2}$ quadric ${\displaystyle {\mathcal {Q}}}$ is called hyperboloid (of one sheet).

Examples:

a) Quadric ${\displaystyle {\mathcal {Q}}}$ in ${\displaystyle P_{2}(K)}$ with form ${\displaystyle \;q({\vec {x}})=x_{1}x_{2}-x_{3}^{2}\;}$ is non-degenerate with index 1.

b) If polynomial ${\displaystyle \;p(\xi )=\xi ^{2}+a_{0}\xi +b_{0}\;}$ is irreducible over ${\displaystyle K}$ the quadratic form ${\displaystyle \;q({\vec {x}})=x_{1}^{2}+a_{0}x_{1}x_{2}+b_{0}x_{2}^{2}-x_{3}x_{4}\;}$ gives rise to a non-degenerate quadric ${\displaystyle {\mathcal {Q}}}$ in ${\displaystyle P_{3}(K)}$ of index 1 (sphere). For example: ${\displaystyle \;p(\xi )=\xi ^{2}+1\;}$ is irreducible over ${\displaystyle \mathbb {R} }$ (but not over ${\displaystyle \mathbb {C} }$ !).

c) In ${\displaystyle P_{3}(K)}$ the quadratic form ${\displaystyle \;q({\vec {x}})=x_{1}x_{2}+x_{3}x_{4}\;}$ 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.^[7][8][9] The reason is the following statement.

A division ring ${\displaystyle K}$ is commutative if and only if any equation ${\displaystyle x^{2}+ax+b=0,\ a,b\in K}$, has at most two solutions.

There are generalizations of quadrics: quadratic sets.^[10] 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.

See also[]

• Klein quadric
• Rotation of axes
• Superquadrics
• Translation of axes

References[]

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", Encyclopedia of Mathematics, EMS Press
• Weisstein, Eric W. "Quadric". MathWorld.

External links[]

• Interactive Java 3D models of all quadric surfaces
• Lecture Note Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes, p. 117