Quadric (projective geometry)

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In mathematics, in the area of projective geometry, a quadric in a projective space is the set of points represented by vectors on which a certain non-trivial quadratic form becomes zero.

1 Definition
2 Basis representation and formula
3 Intersection of lines with quadrics
4 Tangent space and singularity

[edit] Definition

Let K be a field and V a (n+1)-dimensional space ( $n\geq 1$ ) over K. Let F be a non-trivial quadratic form on V. If the point p in PG(n,K) is represented by $v \in V (p=<v>)$ , p is in the quadric Q defined by F if and only if $F (v) = 0$ . Quadrics are well-defined in this way, because all vectors representing a specific point are non-zero multiples of each other and $F (k v) = k 2 F (v)$ due to the definition of a quadratic form.

When n=2, the quadric is also called a conic. When n=3, the quadric is also called a quadratic surface.

[edit] Basis representation and formula

Choosing any basis of V, quadratic forms are exactly those maps from V to K expressible in this way :

$F(v)=\sum_{0\leq i\leq j\leq n}a_{ij}X_{i}X_{j}$

where $(X 0,..., X n)$ are the coordinates of v with respect to the basis. Our restriction that F cannot be trivial comes down to demanding that not all $a i j$ are 0.

An important formula is that for the quadratic form, for all $v=(x_{0},...x_{n}),w=(y_{0},...,y_{n}) \in V, a,b \in K$ :

$F(a v +b w)=a^2 F(v)+b^2 F(w) +a b (x_{0}\frac{\partial F}{\partial X_{0}}(w)+\cdots+ x_{n}\frac{\partial F}{\partial X_{n}}(w))$

Here $\frac{\partial F}{\partial X_{n}}(w)$ stands for the formal derivative :

$\frac{\partial F}{\partial X_{n}}(w)=a_{0i}y_{0}+\cdots+2 a_{ii}y_{i}+\cdots+a_{in}y_{n}$

Plugging in a=b=1 and w=v gives us :

$2 F(v)=x_{0}\frac{\partial F}{\partial X_{0}}(v)+\cdots+ x_{n}\frac{\partial F}{\partial X_{n}}(v)$

[edit] Intersection of lines with quadrics

Examples exist of lines that have no common point with a quadric (as even empty quadrics exists). Let us assume the L contains the point $p=<v> \in Q$ . Let $q = < w >$ be another point on L. All other points on L can be represented for a unique $t\in K$ by $t v + w$ . Searching for other points on L and Q comes down roughly to a linear equation :

$F(t v+w)=0\Longleftrightarrow 0=t^2 0+t (x_{0}\frac{\partial F}{\partial X_{0}}(w)+\cdots+ x_{n}\frac{\partial F}{\partial X_{n}}(w))+F(w)$

$x_{0}\frac{\partial F}{\partial X_{0}}(w)+\cdots+ x_{n}\frac{\partial F}{\partial X_{n}}(w)=0$

we either have no other solutions (if $F(w)\neq 0$ ) or $L\subset Q$ (if $F (w) = 0$ ). If

$x_{0}\frac{\partial F}{\partial X_{0}}(w)+\cdots+ x_{n}\frac{\partial F}{\partial X_{n}}(w)\neq 0$

we have exactly one more common point.

[edit] Tangent space and singularity

Let $p = < v >$ be a point on the quadric Q. We say that $q = < w >$ is in the tangent space of Q at p if q=p or if the line $p q = L$ satisfies :

$L\cap Q=L$ or

p

One denotes the tangent space of Q at p by $T p (Q)$ .

Checking whether q, when different from p, is in the tangent space, comes to down checking if L has no unique other common point with Q, and due to the latter, this means :

$x_{0}\frac{\partial F}{\partial X_{0}}(w)+\ldots+ x_{n}\frac{\partial F}{\partial X_{n}}(w)=y_{0}\frac{\partial F}{\partial X_{0}}(v)+\ldots+ y_{n}\frac{\partial F}{\partial X_{n}}(v)=0$

As 2 F(v)=0, it is obvious that also p itself satisfies this linear equation.

When not all partial derivatives are zero, the equation defines a hyperplane, the tangent hyperplane of Q at p.

When all partial derivatives are zero, the tangent space is the entire space. These points are the singular points of the quadric. A quadric is by definition singular when it has at least one singular point. Singular points can thus be found by solving the equations :