Quadric

In mathematics, a quadric or quadric surface is a generalization of conic sections (ellipses, parabolas, and hyperbolas). In three-dimensional space, quadrics include ellipsoids, paraboloids, and hyperboloids.

More generally, a quadric hypersurface (of dimension D) embedded in a higher dimensional space (of dimension D + 1) is defined as the zero set of an irreducible polynomial of degree two in D + 1 variables; for example, D=1 is the case of conic sections (plane curves). 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.

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 § Normal form of projective quadrics, below.

Formulation

In coordinates x1, x2, ..., xD+1, the general quadric is thus defined by the algebraic equation

∑

i

,

j

=

1

D

+

1

x

i

Q

i

j

x

j

+

∑

i

=

1

D

+

1

P

i

x

i

+

R

=

0

{\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:

x

Q

x

T

+

P

x

T

+

R

=

0

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

where x = (x1, x2, ..., xD+1) is a row vector, xT 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.

Euclidean plane

Quadrics in a Euclidean plane have dimension one and are thus plane curves. They are called conic sections, or conics.