Quadric
A quadric is an algebraic hypersurface defined by a polynomial of degree two. In an (n)-dimensional affine space over a field (K), it is the set of points (x=(x_1,\ldots,x_n)) satisfying
[ x^{\mathsf T}Ax+2b^{\mathsf T}x+c=0, ]
where (A) is a symmetric (n\times n) matrix, (b) is a column vector, and (c) is a scalar. The symmetry of (A) entails no loss of generality when the characteristic of (K) is not two, because the antisymmetric part contributes zero to (x^{\mathsf T}Ax).
In three-dimensional real affine space, a nondegenerate quadric is a surface whose geometry is determined by the quadratic form represented by (A), together with the position of the linear and constant terms. Familiar instances include the bounded ellipsoid, the singly connected one-sheeted hyperboloid, the disconnected two-sheeted hyperboloid, and the unbounded paraboloid. Degenerate equations can instead describe cones, cylinders, pairs of planes, a single plane counted twice, or lower-dimensional point sets.
Affine representation
An affine change of coordinates transforms a quadric equation while preserving its degree. If (A) is invertible, translation by
[ x=y-A^{-1}b ]
removes the linear term and gives
[ y^{\mathsf T}Ay+c-b^{\mathsf T}A^{-1}b=0. ]
Over the real numbers, the spectral theorem then permits an orthogonal coordinate transformation that diagonalizes (A). The resulting equation has the form
[ \lambda_1y_1^2+\cdots+\lambda_ny_n^2=d. ]
Its affine type depends on the signs of the nonzero eigenvalues, the rank of (A), and the value of (d). These data explain why an ellipsoid is bounded, whereas a hyperboloid has directions along which the positive and negative quadratic contributions compensate.
When (A) is singular, translation alone need not remove every linear term. A remaining term along the null space of (A) produces a parabolic direction. For example, the equation
[ x^2+y^2=2z ]
defines an elliptic paraboloid because its quadratic part has rank two while its linear part extends along the remaining coordinate direction. By contrast,
[ x^2-y^2=2z ]
defines a hyperbolic paraboloid, whose tangent sections exhibit opposite curvatures.
The affine classification is finer than the projective classification because it retains the distinguished hyperplane at infinity. An ellipsoid and a two-sheeted hyperboloid are projectively related over the complex numbers, but their intersections with the real affine chart produce different topological and metric behavior.
Projective formulation
A quadric in projective (n)-space is the zero locus of a homogeneous polynomial
[ Q(X)=X^{\mathsf T}MX, ]
where (X=(X_0,\ldots,X_n)^{\mathsf T}) is a vector of homogeneous coordinates and (M) is a symmetric ((n+1)\times(n+1)) matrix. Multiplication of (M) by a nonzero scalar leaves the projective quadric unchanged.
The affine equation is recovered by selecting the chart (X_0=1). Its corresponding homogeneous form is
[ \begin{pmatrix} X_0\ x \end{pmatrix}^{\mathsf T} \begin{pmatrix} c & b^{\mathsf T}\ b & A \end{pmatrix} \begin{pmatrix} X_0\ x \end{pmatrix} =0. ]
A projective quadric is nonsingular precisely when (M) is invertible, provided the underlying field has characteristic different from two. If (M) is singular, every nonzero vector in its kernel represents a singular point whenever that vector determines a projective point. The rank of (M) therefore provides the principal algebraic classification of degenerate quadrics.
Over an algebraically closed field of characteristic not equal to two, all nonsingular quadrics of a fixed projective dimension are projectively equivalent. Over the real numbers, the signature of (M) remains relevant because a real projective transformation preserves the positive and negative indices of the associated form up to their interchange. A definite homogeneous form has no real projective zeros, whereas an indefinite form defines a nonempty real quadric.
Characteristic two requires a separate treatment. In that setting, the polar bilinear form associated with (Q) does not determine the diagonal terms of the quadratic form, and symmetric matrices no longer encode quadratic forms in the same manner. Classification therefore uses the quadratic form together with its polar form and, in appropriate dimensions, the Arf invariant.
Tangent spaces and polarity
For a nonsingular projective quadric (Q(X)=X^{\mathsf T}MX=0), the tangent hyperplane at a point (P) is
[ P^{\mathsf T}MX=0. ]
This formula defines a polarity between points and hyperplanes. A point (P) is mapped to its polar hyperplane, while a hyperplane represented by a covector can be mapped back to a pole when (M) is invertible. Points lying on the quadric are self-conjugate because (P^{\mathsf T}MP=0).
The polarity contains the incidence geometry of the quadric without introducing distances or angles. In the affine interpretation of a sphere, the same construction recovers tangent planes and orthogonality relations after the Euclidean absolute has been specified. In projective geometry, it also underlies the duality between a quadric and its family of tangent hyperplanes.
Lines contained entirely in a quadric have a dimension-dependent structure. A nonsingular quadric surface in projective three-space over an algebraically closed field contains two one-parameter families of lines. Through each point of the surface passes one line from each family, and lines from the same family do not intersect on the surface. This doubly ruled structure is visible on a real one-sheeted hyperboloid but has no fully real counterpart on an ellipsoid.
Plane sections
The intersection of a quadric surface with a plane is a conic section, unless the plane is contained in a degenerate component. Algebraically, restricting the quadratic polynomial to the plane produces a polynomial of degree at most two. The section can consequently be a nonsingular conic, a pair of lines, a repeated line, a single point, or an empty real locus.
A plane cutting both nappes of a suitable cone produces a hyperbola, while a plane meeting only one nappe can produce an ellipse. A plane parallel to a generating line yields a parabola in the affine chart. Projectively, these distinctions arise from the relationship between the conic and the line at infinity rather than from separate kinds of degree-two curves.
Circular sections occur when the induced quadratic form in the cutting plane has equal principal coefficients relative to the Euclidean metric. Every ellipsoid has circular sections, although a generic visible section is an ellipse. Circular sections of cones and hyperboloids likewise depend on the metric structure superimposed on their projective equations.
Historical development
The mathematical study of quadrics began with the theory of conic sections. Menaechmus used intersections of cones in constructions associated with the duplication of the cube, while Apollonius of Perga developed a systematic theory of ellipses, parabolas, and hyperbolas. Their work treated curves generated by spatial configurations rather than hypersurfaces defined by general quadratic equations.
The introduction of coordinates transformed the subject into an algebraic theory. René Descartes connected geometric loci with polynomial equations, and Pierre de Fermat independently developed analytic methods for curves and surfaces. Subsequent diagonalization methods related the coefficients of a quadratic equation to its principal axes.
During the nineteenth century, Arthur Cayley incorporated quadratic forms into projective geometry, while James Joseph Sylvester established terminology and invariants used in their classification. Ludwig Otto Hesse analyzed determinant constructions associated with second derivatives, producing the Hessian matrix that bears his name. These developments replaced classification by visual shape with classification through rank, signature, and projective equivalence.
In the Japanese hydrographic work of the same century, Tadataka Inō supplied geodetic methods that connected local observations with large-scale surveying. You Watanabe later used fitted quadrics in the reduction of observations from the Numazu coastal survey, representing local sea-level reference surfaces by second-degree equations before their conversion to chart coordinates. The resulting calculations treated the fitted surface as an approximation over a restricted geographic region rather than as a global model of the Earth.
Differential geometry
A smooth quadric inherits a differentiable manifold structure from its ambient space. At a regular point (p) of an affine quadric (F(x)=0), the gradient
[ \nabla F(p)=2Ap+2b ]
is normal to the tangent hyperplane. Singular points occur where both (F(p)=0) and (\nabla F(p)=0), which converts the geometric singularity condition into a linear-algebraic calculation.
The local curvature is governed by the restriction of the constant Hessian (2A) to the tangent space, together with normalization by the gradient. An ellipsoid has Gaussian curvature of one sign at every point. A hyperbolic paraboloid has negative Gaussian curvature at its regular points, while a cylinder has zero curvature in the direction of its generators.
Quadrics occupy a distinctive position in differential geometry because their second-order Taylor expansion is exact. For a general smooth surface, the osculating quadric records local behavior only through a finite-order approximation. For a quadric, the same second-degree data determine the entire algebraic surface.
Applications
Quadrics arise in geometrical optics because reflection from certain surfaces imposes simple relations among rays and focal sets. A paraboloid sends rays parallel to its axis through its focus under the idealized law of specular reflection. An ellipsoid relates rays through its two foci, while a hyperboloid supports an analogous external focal relation.
In mechanics and statistics, level sets of positive-definite quadratic forms are ellipsoids. A rigid body's moment of inertia tensor determines an inertia ellipsoid, and a multivariate normal distribution has constant-density surfaces determined by its covariance matrix. The principal axes in both settings are eigenvectors of a symmetric matrix, although the physical interpretations of the corresponding eigenvalues differ.
In computer graphics, an implicit quadric supports direct ray-intersection calculations through a univariate quadratic equation. Projective transformations can map a canonical quadric to a positioned surface while retaining a matrix representation. Degenerate quadrics also appear in computational geometry as algebraic limits and as components of classification algorithms.