Quadrics

Quadrics are algebraic varieties defined by second‑degree polynomial equations. In affine n-space, a quadric is the zero set of a quadratic form q(x) = x^T A x + b^T x + c, where x ∈ R^n, A is a symmetric matrix, b ∈ R^n, and c ∈ R. In projective space P^n, quadrics are given by homogeneous quadratic forms Q(X) = X^T M X, with X = (X0, X1, ..., Xn)^T and M a symmetric matrix; the projective quadric is the set {X : Q(X) = 0}.

The geometry of a quadric is governed by the rank and signature of the quadratic form. Nondegenerate quadrics, corresponding to full-rank forms, are smooth. Degenerate quadrics split into simpler pieces such as pairs of planes, cylinders, or cones.

In two dimensions, quadrics are conic sections. After a suitable linear change of coordinates, a real conic can be classified as an ellipse (including the circle), a parabola, or a hyperbola; the classification reflects the signs of the quadratic part and the presence or absence of linear terms.

In three dimensions, common quadrics include the sphere and ellipsoid (positive-definite quadratic parts), the hyperboloid of one or two sheets, the cone, and the paraboloid. Degenerate 3D quadrics include cylindrical surfaces and unions of planes.

Quadrics are central in many areas: they provide explicit models in differential and algebraic geometry, serve as fundamental objects in computer graphics and CAD, and arise in optimization through quadratic forms. Under linear changes of coordinates, their classification reduces to the congruence class of the quadratic form, enabling classification up to affine or projective equivalence.