Quadric (projective geometry)

In projective geometry a quadric is the set of points of a projective space where a certain quadratic form on the homogeneous coordinates becomes zero. It may also be defined as the set of all points that lie on their dual hyperplanes, under some projective duality of the space.

Formal definition

More formally, let $V$ be an $(n+1)$-dimensional vector space with scalar field $K$, and let $F$ be a quadratic form on $V$. Let $P$ be the $n$-dimensional projective space corresponding to $V$, that is the set $\{ [v]\; :\; v \in V \setminus \{0\} \}$, where $[v]$ denotes the set of all nonzero multiples of $v$. The (projective) quadric defined by $F$ is the set of all points $[v]$ of $P$ such that $F(v)=0$. (This definition is consistent because $[u] = [v]$ implies $u = k v$ for some $k \in K$, and $F(k v)=k^2 F(v)$ by definition of a quadratic form.)

When $P$ is the real or complex projective plane, the quadric is also called a (projective) quadratic curve, conic section, or just conic.

When $P$ is the real or complex projective space, the quadric is also called a (projective) quadratic surface.

In general, if $K$ is the field of real numbers, a quadric is an $(n-1)$-dimensional sub-manifold of the projective space $P$. The exceptions are certain degenerate quadrics that are associated to quadratic forms with special properties. For instance, if $F$ is the trivial or null form (that yields 0 for any vector $v$), the quadric consists of all points of $P$; if $F$ is a definite form (everywhere positive, or everywhere negative), the quadric is empty; if $F$ factors into the product of two non-trivial linear forms, the quadric is the union of two hyperplanes; and so on. Some authors may define "quadric" so as to exclude some or all of these special cases.

Matrix representation

Any quadratic form $F$ can be expressed as

$F(v)=\sum_{0\leq i, j\leq n}M_{i j}v_{i}v_{j}$

where $(v_{0},...,v_{n})$ are the coordinates of $v$ with respect to some chosen basis, and $M$ is a certain $(n+1)\times(n+1)$ symmetric matrix with entries $M_{i j}$ in $K$, that depends on $F$ and on the basis.

This formula can also be written as $F(v)=v\cdot F'(v)$ where $\cdot$ is the standard inner product of $K^{n+1}$, and $F'(v)$ is the vector of $K^{n+1}$ defined by

$(F'(v))_{i}=M_{i 0}v_{0}+ \cdots +M_{i n}v_{n}$

The quadratic form $F$ is trivial if and only if all the entries $M_{i j}$ are 0. If $K$ is the real numbers, there is always a basis such that $M$ is a diagonal matrix. In this case, the signs of the diagonal elements $M_{i i}$ determine whether the quadric is degenerate or not.

Polarity, tangent hyperplane, and singular points

In general, a projective quadric $Q$ defines a projective polarity: a mapping that takes any point $p=[v]$ of $P$ to a hyperplane $p^\ast$ of $P$, and vice-versa, while preserving the incidence relation between points and hyperplanes. The coefficient vector of the polar hyperplane $h = p^\ast$, relative to the chosen basis of $V$, is $F'(v)$.

If $p$ is not on the quadric, the hyperplane $h = p^\ast$ is well-defined (that is, not identically zero) and does not contain $p$.

If $p$ is on the quadric and the hyperplane $h = p^\ast$ is well-defined, and contains $p$ (which is said to be a regular point). In fact, it is the hyperplane that is tangent to the quadric at $p$.

If $p$ is on the quadric, it may happen that all coefficients $h_{i}$ are zero. In that case the polar $p^\ast$ is not defined, and $p$ is said to be a singular point or singularity of the quadric.

The tangent hyperplane turns out to be the union of all lines that are either entirely contained in $Q$, or intersect $Q$ at only one point.

The condition for a point $[u]$ to be in the hyperplane that is tangent to $Q$ at $[v]$ is $u \cdot F'(v) = 0$, which is equivalent to $v \cdot F'(u) = 0$

The condition for a point $[v]$ to be singular is $F'(v) = 0$. The quadric has singular points if and only the matrix $M$, in diagonal form, has one or more zeros in its diagonal. It follows that the set of all singular points on the quadric is a projective subspace.

Intersection of lines with quadrics

In projective space, a straight line may intersect a quadric $Q$ at zero, one, or two points, or may be entirely contained in it. The line defined by two distinct points $p=[u]$ and $q=[v]$ is the set of points of the form $[a u + b v]$ where $a,b$ are arbitrary scalars from $K$, not both zero. This generic point lies on $Q$ if and only if $F(a u + b v)=0$, which is equivalent to

$a^2 F(u) + a b (u \cdot (\partial F(v))) + b^2 F(v) = 0$

The number of intersections depends on the three coefficients $A = F(u)$, $B = u \cdot (\partial F(v))$, and $C = F(v)$. If all three of $A,B,C$ are zero, any pair $a,b$ satisfies the equation, so the line is entirely contained in $Q$. Otherwise, the line has zero, one, or two distinct intersections depending on whether $\Delta = B^2 - 4 A C$ is negative, zero, or positive, respectively.