Quadric

In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in (D + 1)-dimensional space defined as the locus of zeros of a quadratic polynomial. In coordinates x₁, x₂, ..., x_D+1, the general quadric is defined by the algebraic equation

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

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 ring. In general, the locus of zeros of a set of polynomials is known as an algebraic set, and is studied in the branch of algebraic geometry.

A quadric is thus an example of an algebraic set. For the projective theory see Quadric (projective geometry).

Euclidean plane and space

Quadrics in the Euclidean plane are those of dimension D = 1, which is to say that they are curves. Such quadrics are the same as conic sections, and are typically known as conics rather than quadrics.

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. We shall restrict ourself to the case of finite-dimensional projective spaces.

Quadratic forms

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

is called quadratic form. The bilinear form is symmetric.

In case of we have , i.e. and are mutually determined in a unique way.
In case of we have always , i.e. is symplectic.

For and ( is a base of ) has the form

For example:

Definition and properties of a quadric

Below let be a field, , and the n-dimensional projective space over , i.e.

the set of points. ( is a (n + 1)-dimensional vector space over the field and is the 1-dimensional subspace generated by ),

the set of lines.

Additionally let be a quadratic form on vector space . A point is called singular if . The set

of singular points of is called quadric (with respect to the quadratic form ). For point the set