QuadricEdit

A quadric is a surface in three-dimensional space defined by a second-degree polynomial equation in three variables. In its most general form, a quadric in coordinates (x, y, z) satisfies an equation of the form

ax^2 + by^2 + cz^2 + dxy + eyz + fzx + gx + hy + iz + j = 0,

where the coefficients a, b, c, d, e, f, g, h, i, j are real numbers. The quadratic terms determine the overall curvature and orientation, while the linear terms and constant term translate and scale the surface. When cross terms (like dxy, eyz, fzx) are present, the quadric may be rotated relative to the coordinate axes; a suitable rotation can often remove the cross terms to reveal the surface’s principal axes. Quadrics encompass a broad family of familiar shapes, including spheres, ellipsoids, paraboloids, hyperboloids, cones, and cylinders, each arising from specific choices of coefficients and, in some cases, from translations of a simpler, centered form. They are central objects in analytic geometry, linking algebraic equations to geometric surfaces, and they arise in optics, architecture, physics, computer graphics, and many other fields. See conic section and analytic geometry for related ideas.

Definition and notation

A quadric surface can be described more compactly using a symmetric 3×3 matrix A, a vector b for the linear terms, and a scalar c for the constant term. Writing x for the column vector (x, y, z)^T, the quadric is the zero set of

q(x) = x^T A x + b^T x + c.

Because A is symmetric, it encodes the curvature of the surface, while b and c translate or rescale it. By translating the coordinate system to remove linear terms and by rotating to diagonalize A (orbital or principal-axis transformations), one can bring many quadrics to simpler, canonical forms. The classification of quadrics often hinges on the eigenvalues (the spectrum) of A and the rank (which relates to degeneracy). See Sylvester's law of inertia and quadratic form for the underlying linear-algebra ideas that explain why different signatures correspond to ellipsoidal, hyperbolic, or parabolic families.

Canonical forms and classification

Real quadrics in three variables fall into several broad families, distinguished by the signs and magnitudes of the eigenvalues of A and by whether the surface is centered or translated. After completing the square and applying an orthogonal change of variables, many quadrics can be written in forms such as:

Ellipsoid: x′^2/a^2 + y′^2/b^2 + z′^2/c^2 = 1, with a, b, c > 0. See Ellipsoid.
Sphere: x^2 + y^2 + z^2 = r^2, a special case of an ellipsoid with a = b = c. See Sphere.
Elliptic paraboloid: z′ = x′^2/a^2 + y′^2/b^2, opening in the positive z′ direction. See Elliptic paraboloid.
Hyperbolic paraboloid: z′ = x′^2/a^2 − y′^2/b^2, a saddle-shaped surface. See Hyperbolic paraboloid.
Hyperboloid of one sheet: x′^2/a^2 + y′^2/b^2 − z′^2/c^2 = 1, a connected surface with a waist along the middle. See Hyperboloid of one sheet.
Hyperboloid of two sheets: −x′^2/a^2 − y′^2/b^2 + z′^2/c^2 = 1, consisting of two disjoint components along the z′ axis. See Hyperboloid of two sheets.
Cone (quadratic cone): x′^2/a^2 + y′^2/b^2 − z′^2/c^2 = 0, the limiting case between elliptic and hyperbolic types.
Elliptic cylinder: x′^2/a^2 + y′^2/b^2 = 1 with z free, a surface extending indefinitely in the z direction. See Elliptic cylinder.
Parabolic cylinders and other degenerate cases: examples include surfaces that separate into planes or lines when certain coefficients vanish. See Degenerate quadric.

Cross terms in the original equation correspond to a rotation of axes; after such a rotation, the surface can often be read off from the diagonal form. The overall type is determined by the signature (the counts of positive, negative, and zero eigenvalues) of A and by whether the surface is bounded or unbounded.

Common quadric surfaces

Sphere and ellipsoid: highly symmetric, with all normal curvatures of the same sign. In many practical problems, spheres and ellipsoids model symmetric bodies, lenses, and optical components. See Sphere and Ellipsoid.
Paraboloids: surfaces where one coordinate is a quadratic function of the other two. Elliptic paraboloids occur in reflectors and architectural shells, while hyperbolic paraboloids (saddles) arise in cooling towers and modern design. See Paraboloid and Hyperbolic paraboloid.
Hyperboloids: one-sheet and two-sheet variants appear in architecture (e.g., cooling towers, cooling towers with hyperbolic shapes) and in certain physical models where one direction behaves differently from the others. See Hyperboloid of one sheet and Hyperboloid of two sheets.
Cones: the quadric cone is the simplest nontrivial surface through the origin, important in perspective drawing and in describing asymptotic directions on other quadrics. See Cone (geometry).
Cylinders: finite- or infinite-length surfaces with a constant cross-section (such as a circle) extruded along a third direction. See Elliptic cylinder and Cylindrical surface.

Cross-sections and projections

Intersecting a quadric with a plane typically yields a conic section: an ellipse, a parabola, or a hyperbola. This two-dimensional family of curves generalizes to the three-dimensional quadric setting, and vice versa. Projections of quadrics under linear maps preserve their quadratic nature in many cases, providing a bridge between 3D geometry and 2D sketches or computer-generated images. See Conic section and Projective geometry.

Algebraic and geometric properties

Degeneracy: when the determinant of A is zero, a quadric may break into a union of planes, a pair of lines, or a single line with multiplicity; these are called degenerate quadrics. See Degenerate quadric.
Invariants and normalization: the classification relies on invariants such as the rank of A and the signature, which remain under appropriate linear changes of coordinates. See Invariant (mathematics).
Real vs complex: over the complex numbers, quadrics admit a more uniform diagonalization, but real quadrics exhibit the familiar three-dimensional shapes with real cross sections and real loci. See Real algebraic geometry.

Applications

Architecture and design: the shapes of domes, shells, and canopies often exploit quadrics for structural efficiency and aesthetic clarity. See Architectural geometry.
Optics and reflectors: parabolic and elliptical quadrics model focusing properties of mirrors and dishes in telescopes and communication systems. See Parabolic reflector and Elliptical mirror.
Computer graphics and CAD: implicit surface representations include quadric equations; fast ray tracing and collision detection benefit from the well-understood algebraic form. See Computer-aided design and Implicit surface.
Physics and spacetime models: certain quadratic forms encode metric properties in both classical and relativistic contexts; the idea of a quadric appears in discussions of light cones and spacetime diagrams. See Minkowski space and Quadratic form.

History

The classical study of curves and surfaces traces back to ancient geometry for conic sections and to the emergence of analytic geometry in the early modern period. The systematic treatment of quadrics as second-degree surfaces developed during the 17th through 19th centuries with contributions from Descartes, Apollonius of Perga (through the lineage that led to conic sections), and later figures in the development of analytic geometry and linear algebra. The modern algebraic approach emphasizes the quadratic form, matrix representation, and canonical forms, linking geometry with Linear algebra and Matrix (mathematics).