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Quadric

In mathematics, a quadric is a hypersurface defined by a quadratic polynomial equation. In three-dimensional space, it is a second-order algebraic surface given by the general equation ax^2 + by^2 + cz^2 + dxy + exz + fyz + gx + hy + iz + j = 0, where the coefficients determine the specific shape.^[1] These surfaces generalize conic sections—such as ellipses, parabolas, and hyperbolas—from two dimensions to three, encompassing a variety of geometric forms including ellipsoids, hyperboloids, paraboloids, cones, and cylinders.^[2] Quadrics are fundamental in algebraic geometry and calculus, appearing in applications ranging from computer graphics and physics to optimization problems, due to their smooth, curved structures that model phenomena like planetary orbits or light reflection.^[3] The classification of quadrics relies on reducing the general equation to one of 17 standard canonical forms through coordinate transformations, which reveal their symmetries and properties such as ellipticity, hyperbolicity, or degeneracy.^[1] Non-degenerate quadrics, like the ellipsoid \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 or the hyperboloid of one sheet \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1, are bounded or unbounded depending on the signs of the quadratic terms, while degenerate cases include pairs of planes or single planes.^[4] In projective geometry, quadrics extend to higher dimensions as hypersurfaces defined by homogeneous quadratic polynomials, unifying their study across Euclidean, affine, and projective spaces.^[1] Quadric surfaces exhibit traces—cross-sections parallel to the coordinate planes—that are conic sections, aiding in their identification and visualization; for instance, an elliptic paraboloid z = x^2 + y^2 yields ellipses in horizontal planes and parabolas in vertical planes.^[3] Their mathematical significance stems from quadratic forms, where the associated symmetric matrix's eigenvalues dictate the surface's type: positive definite for ellipsoids, indefinite for hyperboloids.^[2] The mathematical study of quadrics developed in the 17th century with the introduction of analytic geometry, evolving into tools for modern fields such as computer-aided design and general relativity.^[5]

Fundamentals of Quadrics

Definition and Equation

A quadric hypersurface is defined as the set of points in projective space \mathbb{P}^n where a homogeneous quadratic polynomial in n+1 variables equals zero.^[6] This algebraic variety, of dimension n-1 and degree two, arises as the zero locus X = V(Q) of a single such polynomial Q.^[6] In affine space \mathbb{A}^n, the general equation of a quadric takes the form

\sum_{i=1}^n \sum_{j=1}^n a_{ij} x_i x_j + \sum_{i=1}^n b_i x_i + c = 0,

where the quadratic terms are present (i.e., not all a_{ij} = 0), incorporating cross terms like $2a_{12}xy in lower dimensions.^[7] To embed this in projective space \mathbb{P}^n, homogenize the equation using an additional variable x_0, yielding the form

\sum_{0 \leq i \leq j \leq n} a_{ij} x_i x_j = 0,

a homogeneous quadratic polynomial of degree two.^[6] This equation corresponds to a quadratic form Q(\mathbf{x}) = \mathbf{x}^T A \mathbf{x}, where \mathbf{x} = (x_0, \dots, x_n)^T is the coordinate vector and A = (a_{ij}) is a symmetric (n+1) \times (n+1) matrix, ensuring the form is well-defined under scalar multiplication in projective coordinates.^[8] Quadrics are degenerate when the matrix A is singular (i.e., \det(A) = 0), in which case the hypersurface factors into lower-degree components, such as lines or planes in projective space.^[7]

Basic Properties in Euclidean Space

In Euclidean space \mathbb{R}^n, a quadric hypersurface is defined as the level set \{x \in \mathbb{R}^n \mid Q(x) = c\}, where Q(x) = x^T A x + b^T x + d is a quadratic function with symmetric matrix A, linear term b, and constant d.^[9] The geometric properties depend on the eigenvalues of A and the value of c. For positive definite A (all eigenvalues positive), the level set for appropriate c > 0 forms an ellipsoid, which is compact and bounded.^[9] In contrast, indefinite A (mixed positive and negative eigenvalues) yields hyperboloids, which are unbounded.^[9] Elliptic quadrics, such as ellipsoids arising from positive definite quadratic forms, bound convex regions, meaning the line segment joining any two points on the surface lies within the bounded region they enclose.^[10] Hyperbolic quadrics, like hyperboloids from indefinite forms, are non-convex, as they feature saddle-like regions where line segments between points may exit the surface.^[9] These convexity distinctions follow from the signature of the quadratic form, with positive definite cases ensuring the sublevel sets \{x \mid Q(x) \leq c\} are convex bodies.^[10] Non-degenerate quadrics, where A has full rank (determinant nonzero), form smooth hypersurfaces away from singular points, as the gradient \nabla Q(x) = 2Ax + b vanishes only at isolated vertices or along degenerate directions, making the level set a smooth (n-1)-manifold.^[9] For surfaces in \mathbb{R}^3, the Gaussian curvature K, defined as the product of principal curvatures, characterizes local geometry: ellipsoids have positive K > 0 everywhere, indicating elliptic points; one-sheet hyperboloids have negative K < 0, marking hyperbolic points; and two-sheet hyperboloids have positive K > 0 on each sheet.^[11] Under orthogonal transformations O \in O(n), which preserve the Euclidean inner product \langle x, y \rangle, quadrics transform to quadrics of the same type, as Q(Ox) = x^T (O^T A O) x + (O^T b)^T x + d yields a congruent symmetric matrix O^T A O with unchanged eigenvalues and signature.^[9] This invariance ensures that properties like compactness, convexity, and curvature signs are preserved, facilitating canonical forms via diagonalization in an orthonormal basis.^[9]

Quadrics in Low Dimensions

Quadrics in the Euclidean Plane

In the Euclidean plane, quadrics are represented by conic sections, which are the curves obtained by intersecting a plane with a right circular cone.^[12] The general equation of a conic section is ax^2 + bxy + cy^2 + dx + ey + f = 0, where a, b, c, d, e, f are real coefficients, and not all of a, b, c are zero.^[12] The classification of non-degenerate conics depends on the discriminant \Delta = b^2 - 4ac. If \Delta < 0, the conic is an ellipse (including a circle as a special case when a = c and b = 0); if \Delta = 0, it is a parabola; and if \Delta > 0, it is a hyperbola.^[12] Standard equations for these conics, assuming centered or vertex-aligned forms without rotation, are as follows. For an ellipse centered at the origin with semi-major axis a and semi-minor axis b (where a > b > 0):

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

For a parabola with vertex at the origin opening to the right, with focal length p > 0:

y^2 = 4px

For a hyperbola centered at the origin with transverse axis along x of semi-length a > 0 and conjugate axis semi-length b > 0:

\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

^[13]^[14]^[15] Degenerate conics arise when the quadratic form factors into linear terms or represents trivial loci, resulting in four types: a single point (e.g., x^2 + y^2 = 0), two intersecting lines (e.g., xy = 0), two parallel lines (e.g., x^2 - 1 = 0), or the empty set (e.g., x^2 + y^2 + 1 = 0).^[12] Each non-degenerate conic is characterized by geometric properties involving foci, directrix, and eccentricity e, defined as the ratio of the distance from a point on the conic to a focus over the distance to the corresponding directrix. For an ellipse, there are two foci at (\pm c, 0) where c = \sqrt{a^2 - b^2}, directrices at x = \pm \frac{a}{e}, and $0 \leq e < 1. For a parabola, the single focus is at (p, 0), the directrix is x = -p, and e = 1. For a hyperbola, the foci are at (\pm c, 0) where c = \sqrt{a^2 + b^2}, directrices at x = \pm \frac{a}{e}, and e > 1.^[13]^[14]^[15] A line intersects a conic section in at most two points, as substituting the line equation into the conic yields a quadratic equation in one variable with up to two real roots.^[12]

Quadrics in Euclidean Three-Space

In Euclidean three-space, quadric surfaces are the loci of points satisfying a second-degree equation in three variables, generalizing conic sections to three dimensions. These surfaces encompass a variety of shapes, including bounded and unbounded forms, and play a fundamental role in geometry due to their symmetry and intersection properties. Unlike their planar counterparts, quadric surfaces exhibit volumetric characteristics and can extend infinitely in certain directions./12%3A_Vectors_in_Space/12.06%3A_Quadric_Surfaces) The classification of non-degenerate quadric surfaces in Euclidean three-space includes ellipsoids, hyperboloids of one or two sheets, elliptic paraboloids, and hyperbolic paraboloids. Degenerate cases comprise cones and cylinders. Ellipsoids are closed, bounded surfaces resembling stretched spheres, while hyperboloids of one sheet form connected, saddle-like structures that extend infinitely. Hyperboloids of two sheets consist of two separate unbounded components. Elliptic paraboloids are bowl-shaped and open in one direction, whereas hyperbolic paraboloids resemble saddles with rulings in two directions. Cones taper to a vertex and extend infinitely, and cylinders are extruded conics along a line, lacking a vertex.^[2]^[1] Standard equations illustrate these forms; for an ellipsoid aligned with the coordinate axes, the equation is

\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1,

where a, b, c > 0 determine the semi-axes lengths. For a hyperboloid of one sheet, it is

\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1,

with a, b, c > 0, yielding a surface that narrows at the waist and flares outward. These equations assume principal axis alignment, but general quadrics involve cross terms resolved via orthogonal transformations./12%3A_Vectors_in_Space/12.06%3A_Quadric_Surfaces)^[2] Key properties distinguish these surfaces. Ellipsoids are compact and convex, enclosing a finite volume. Hyperboloids of one sheet and two sheets, along with cones, exhibit asymptotic behavior: hyperboloids approach a conical surface at infinity, known as their asymptotic cone, beyond which the surface does not extend. Hyperboloids of one sheet and cones are ruled surfaces, containing families of straight lines (rulings) lying entirely on the surface; the former is doubly ruled with two such families. Hyperbolic paraboloids are also doubly ruled, facilitating applications in architecture like minimal surfaces. Cylinders inherit the properties of their generating conic, remaining parallel and unbounded.^[16]^[17]^[18] Visualization of quadric surfaces often relies on their cross-sections, or traces, in coordinate planes, which yield conic sections such as ellipses, hyperbolas, parabolas, or degenerate forms. For instance, horizontal slices of an ellipsoid produce ellipses, while vertical slices of a hyperboloid of one sheet yield hyperbolas. These planar intersections provide a direct link to two-dimensional conics, aiding in identification and sketching./12%3A_Vectors_in_Space/12.06%3A_Quadric_Surfaces)^[2] Over the real numbers, some quadrics may lack real points or consist partly of imaginary components, such as the imaginary ellipsoid x^2 + y^2 + z^2 = -1, which has no real points but serves as a precursor to projective geometry where complex extensions unify real and imaginary cases.^[19]

Projective Quadrics

Formulation in Projective Space

In projective geometry, the real projective space \mathbb{RP}^n is defined as the set of lines through the origin in \mathbb{R}^{n+1}, or equivalently, the quotient space \mathbb{R}^{n+1} \setminus \{\mathbf{0}\} / \sim where \mathbf{x} \sim \lambda \mathbf{x} for \lambda \in \mathbb{R} \setminus \{0\}.^[20] Points in \mathbb{RP}^n are represented using homogeneous coordinates [x_1 : \cdots : x_{n+1}], where the coordinates are defined up to nonzero scalar multiplication, allowing a uniform treatment of affine points and directions at infinity.^[20] A projective quadric in \mathbb{RP}^n is the zero set of a homogeneous quadratic form Q(\mathbf{x}) = \mathbf{x}^T A \mathbf{x}, where A is a symmetric (n+1) \times (n+1) matrix and \mathbf{x} = (x_1, \dots, x_{n+1})^T \in \mathbb{R}^{n+1}.^[21] This defines a hypersurface consisting of all points [ \mathbf{x} ] \in \mathbb{RP}^n such that Q(\mathbf{x}) = 0, with the homogeneity ensuring that the equation is well-defined under scaling of coordinates.^[20] The connection to affine quadrics arises through dehomogenization: in the affine chart where x_{n+1} \neq 0, setting x_{n+1} = 1 yields affine coordinates (u_1, \dots, u_n) = (x_1/x_{n+1}, \dots, x_n/x_{n+1}), transforming the projective equation into a nonhomogeneous quadratic in \mathbb{R}^n.^[20] Conversely, any affine quadric in \mathbb{R}^n, given by a quadratic equation in these coordinates, homogenizes to a projective quadric by introducing the extra variable x_{n+1} and multiplying linear and constant terms appropriately.^[21] This projective completion incorporates points at infinity, corresponding to the hyperplane x_{n+1} = 0 in homogeneous coordinates, which "closes" the affine quadric into a compact hypersurface.^[20] For instance, the affine ellipse x^2/a^2 + y^2/b^2 = 1 in \mathbb{R}^2 homogenizes to the projective conic x^2/a^2 + y^2/b^2 - z^2 = 0 in \mathbb{RP}^2, where the points at infinity (z=0) form the line at infinity intersecting the conic at two points.^[21] In the projective plane \mathbb{RP}^2, Bézout's theorem provides a basic result on intersections: two curves of degrees m and n with no common components intersect in exactly mn points, counting multiplicities and including points at infinity.^[22] For two projective conics (degree 2 curves), this implies they intersect in 4 points, offering a foundational tool for understanding their geometric interactions.^[22]

Normal Forms and Classification

In projective space over the real numbers, the classification of quadrics relies on reducing the associated quadratic form to a normal form through linear transformations, specifically congruence transformations that preserve the projective structure. A quadratic form Q(\mathbf{x}) = \mathbf{x}^T A \mathbf{x}, where A is a real symmetric matrix, defines the quadric as the set \{\mathbf{x} \in \mathbb{RP}^n \mid Q(\mathbf{x}) = 0 \}. By Sylvester's law of inertia, any such form is congruent to a diagonal matrix with p entries of +1, q entries of -1, and r = (n+1) - p - q zeros, where the numbers p, q, and r are invariants under congruence transformations A \mapsto P^T A P for invertible P \in \mathrm{GL}(n+1, \mathbb{R}).^[23]^[21] This diagonalization yields the normal form

Q(\mathbf{x}) = \sum_{i=1}^p x_i^2 - \sum_{j=1}^q x_{p+j}^2 = 0,

where the variables corresponding to zero eigenvalues can be set to zero, effectively reducing to a quadric in a projective subspace of dimension p + q - 1. The rank of the quadric is k = p + q, measuring its degeneracy (degenerate if k < n+1), and the signature is s = p - q, which distinguishes the geometric type. Non-degenerate quadrics have full rank k = n+1 and r = 0.^[21] Quadrics are classified by their rank and signature. Degenerate cases include pairs of hyperplanes (k = 2) or a double hyperplane (k = 1), but non-degenerate quadrics fall into hyperbolic and elliptic types based on the signature. Hyperbolic quadrics have balanced signature |s| \leq 2 (often p = q or p = q+1), featuring real rulings and hyperbolic cross-sections, such as the hyperboloid of one sheet given by x_1^2 + x_2^2 - x_3^2 - x_4^2 = 0 in \mathbb{RP}^3. Elliptic quadrics have definite or nearly definite signatures (e.g., p = n+1, q = 0), often empty over the reals, like the imaginary ellipsoid x_1^2 + x_2^2 + x_3^2 + x_4^2 = 0 in \mathbb{RP}^3, though they may contain real points in lower-dimensional slices. Parabolic types arise in affine views but are unified projectively via the signature.^[21] For conics in the real projective plane \mathbb{RP}^2 (dimension n=2, rank 3), non-degenerate cases have signatures (3,0), (0,3) (empty ovals, no real points), or (2,1) (with real points). The projective classification distinguishes ovals (bounded components, like ellipses, intersecting the line at infinity in 0 points), hyperbolas (two components or unbounded, intersecting in 2 points), and parabolas (unbounded with one vertex, tangent to the line at infinity in 1 point). These distinctions emerge when embedding the projective conic into an affine plane by choosing a line at infinity, with the intersection multiplicity determining the type: 0 for oval/ellipse, 1 for parabola, and 2 for hyperbola. All non-degenerate conics with real points are projectively equivalent over \mathbb{R}, but their affine realizations differ topologically.^[24]

Parametrization and Points

Rational Parametrization

A rational parametrization of a quadric provides a birational map from an affine or projective space to the quadric surface, allowing points on the quadric to be expressed using rational functions of parameters. This approach is particularly useful for non-degenerate quadrics, as it establishes that such varieties are rational over fields like the reals, meaning they are birationally equivalent to projective space of one dimension less. The stereographic projection serves as a fundamental method to construct these parametrizations by projecting from a chosen point on the quadric to a complementary line or plane, yielding explicit rational coordinates.^[25]^[21] For conics in the plane, such as circles, the stereographic projection from a point on the conic to a line provides a birational equivalence to the projective line \mathbb{P}^1. Consider the unit circle defined by x^2 + y^2 = 1 in the affine plane, which can be homogenized to x_0^2 + x_1^2 = x_2^2 in \mathbb{P}^2. Projecting from the point [1, 0, 1] on the conic to the line x_0 = 0 yields the parametrization:

[x : y : z] = [1 - t^2 : 2t : 1 + t^2],

where t \in \mathbb{R} parameterizes the line via [0 : 1 : t]. Dehomogenizing by setting z = 1 gives the affine map (x, y) = \left( \frac{1 - t^2}{1 + t^2}, \frac{2t}{1 + t^2} \right), which covers all points except the projection point and is rational over the reals. This construction works for any non-degenerate conic over the reals, as the existence of a real point ensures the projection is defined.^[21]^[26] In three-dimensional Euclidean space, stereographic projection extends to quadric surfaces, mapping from a point on the surface to the plane tangent at the opposite pole, establishing birational equivalence to \mathbb{P}^2. For the unit sphere x^2 + y^2 + z^2 = 1, projecting from the north pole (0, 0, 1) to the equatorial plane z = 0 gives the inverse map (u, v) = \left( \frac{x}{1 - z}, \frac{y}{1 - z} \right), and the forward parametrization:

(x, y, z) = \left( \frac{2u}{1 + u^2 + v^2}, \frac{2v}{1 + u^2 + v^2}, \frac{1 - u^2 - v^2}{1 + u^2 + v^2} \right),

which is rational over the reals and bijective except at the pole. This method applies to any non-degenerate quadric surface over the reals, provided it is smooth, as the projection avoids singularities and covers the surface densely.^[26] Ellipsoids, as affine transformations of spheres, inherit rational parametrizations by scaling the spherical map. For the ellipsoid \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 with a, b, c > 0, projecting from the pole (0, 0, c) to the plane z = 0 yields:

(x, y, z) = \left( \frac{2a^2 b^2 u}{b^2 u^2 + a^2 v^2 + a^2 b^2}, \frac{2a^2 b^2 v}{b^2 u^2 + a^2 v^2 + a^2 b^2}, \frac{c (b^2 u^2 + a^2 v^2 - a^2 b^2)}{b^2 u^2 + a^2 v^2 + a^2 b^2} \right),

a rational map over the reals that parametrizes the surface birationally. Over the rationals, such parametrizations exist if the quadric has a rational point, enabling the projection to be defined with rational coefficients; otherwise, they may require extensions.^[26]^[25]

Rational Points on Quadrics

A rational point on a quadric is a point with coordinates in the rational numbers \mathbb{Q} that satisfies the defining quadratic equation.^[27] The existence of rational points on a quadric over \mathbb{Q} is governed by the Hasse-Minkowski theorem, which asserts that a quadratic form over \mathbb{Q} represents zero non-trivially if and only if it does so over the real numbers \mathbb{R} and over the p-adic rationals \mathbb{Q}_p for every prime p. This local-global principle provides a complete criterion for the solubility of the corresponding projective quadric over \mathbb{Q}.^[28] If a quadric over \mathbb{Q} has at least one rational point, it is birationally equivalent over \mathbb{Q} to projective space, allowing a rational parametrization of all its rational points. This parametrization implies that the set of rational points is infinite. Moreover, since the parameters can be chosen rationally and rationals are dense in the reals, the rational points are dense in the real points of the quadric.^[27] A classic example arises on the quadric x^2 + y^2 = z^2 over \mathbb{Q}, whose rational points correspond to Pythagorean triples (x, y, z) with x, y, z \in \mathbb{Q} and \gcd(x, y, z) = 1. These points are generated by the formulas x = m^2 - n^2, y = 2mn, z = m^2 + n^2 for coprime integers m > n > 0 of opposite parity. This parametrization produces all primitive Pythagorean triples upon clearing denominators.^[29] Hilbert's irreducibility theorem ensures the existence of infinitely many specializations of a general quadric over \mathbb{Q}(t) that retain rational points over \mathbb{Q}, contributing to the abundance of rational points across families of quadrics.^[30]

Quadrics over General Fields

Quadratic Forms and Projective Quadrics

A quadratic form over a field F is a homogeneous polynomial Q: V \to F of degree 2 on a finite-dimensional vector space V over F, or equivalently, in coordinates, Q(x_1, \dots, x_n) = \sum_{i,j=1}^n a_{ij} x_i x_j with a_{ij} = a_{ji}.^[31] It is associated to a symmetric bilinear form B: V \times V \to F via Q(x) = B(x, x), where B is defined by B(x + y, x + y) - B(x, x) - B(y, y) = 2B(x, y) when \mathrm{char}(F) \neq 2; in characteristic 2, quadratic forms are defined separately without requiring symmetry of B, as the alternation B(x, y) - B(y, x) may not vanish.^[31]^[32] The associated projective quadric is the subvariety of the projective space \mathbb{P}^n(F) consisting of points = [x_1 : \dots : x_{n+1}] such that Q(x) = 0, where V = F^{n+1}.^[33] This defines a projective hypersurface, and for non-constant Q, it is a quadric variety over F. A quadratic form Q is non-degenerate if the associated bilinear form B has full rank, meaning the matrix of B has non-zero determinant or, equivalently, for every nonzero v \in V, the linear functional w \mapsto B(v, w) is nonzero.^[31] Non-degenerate quadrics correspond to smooth hypersurfaces in projective space, without singular points. Over any field F, a non-degenerate quadratic space (V, Q) admits a Witt decomposition V \cong \bigoplus_{i=1}^r H \oplus A, where each H is a hyperbolic plane (2-dimensional space with Witt index 1, spanned by isotropic vectors with B(e, f) = 1), r is the Witt index (dimension of a maximal isotropic subspace), and A is the anisotropic kernel (the unique anisotropic quotient up to isometry, with no nonzero isotropic vectors).^[32] The Witt index measures the maximal dimension of totally isotropic subspaces, and the anisotropic part captures the "indecomposable" core; in characteristic 2, the decomposition holds but hyperbolic planes are defined via alternating forms.^[32] Over finite fields F_q with q elements, the Chevalley-Warning theorem implies that any non-degenerate quadratic form in at least three variables has a nontrivial zero, ensuring the associated projective quadric in \mathbb{P}^n(F_q) with n \geq 2 contains at least one point; moreover, the number of affine points on the cone Q(x) = 0 in F_q^{n+1} is congruent to 0 modulo q, yielding the projective point count as (N - 1)/(q - 1) where N \equiv 0 \pmod{q}.^[34]^[35] For smooth quadrics in \mathbb{P}^{m-1}(F_q), the exact number of points is \frac{q^{m-1} - 1}{q - 1} if the Witt index is (m-1)/2 (even dimension case) or \frac{q^{m-1} - 1}{q - 1} \pm q^{(m-2)/2} otherwise, depending on the type.^[36] This framework generalizes the classification of quadrics over the reals, where anisotropic forms are definite and hyperbolic parts yield indefinite signatures.^[31]

Intersections, Radicals, and Subspaces

In projective space over a general field k, the intersection of a quadric hypersurface Q \subset \mathbb{P}^{n-1}_k, defined by a quadratic form \phi: V \to k on an n-dimensional vector space V, with a line \ell \cong \mathbb{P}^1_k can consist of 0, 1, or 2 points.^[21] This follows from restricting \phi to the 2-dimensional subspace corresponding to \ell, yielding a binary quadratic form whose roots determine the intersection points; over algebraically closed fields, secant lines intersect at two distinct points, while tangent lines touch at one point with multiplicity two.^[21] In general fields, the number of points depends on the splitting of the restricted form, potentially yielding no rational points if anisotropic.^[37] Associated to the quadratic form \phi is its polar bilinear form b_\phi(u,v) = \phi(u+v) - \phi(u) - \phi(v). The f-radical of \phi, denoted \mathrm{rad} \, \phi, is \{v \in V \mid \phi(v) = 0 \textrm{ and } b_\phi(v, w) = 0 \ \forall w \in V\}.^[37] The q-radical, or radical of the bilinear form, is \mathrm{rad} \, b_\phi = \{v \in V \mid b_\phi(v,w) = 0 \ \forall w \in V\}, the orthogonal complement to V under b_\phi.^[37] For nondegenerate \phi (i.e., \mathrm{rad} \, b_\phi = 0), \mathrm{rad} \, \phi = \{0\}, a proper subset of the isotropic cone; in degenerate cases, \mathrm{rad} \, \phi \subseteq \mathrm{rad} \, b_\phi captures the singularity locus.^[37] The index of the quadric Q, also called the Witt index of \phi, is the dimension of a maximal isotropic subspace of V, where an isotropic subspace W \subseteq V satisfies b_\phi|_W \equiv 0 (totally singular for the bilinear form).^[32]^[37] By Witt's decomposition theorem, any quadratic space decomposes as a direct sum of the radical and hyperbolic planes plus an anisotropic kernel, with the index equal to the number of hyperbolic planes (half the dimension of the hyperbolic part).^[32] For a nondegenerate form of even dimension $2m, the index is at most m, achieved when \phi is hyperbolic; in odd dimension $2m+1, it is at most m.^[37] A q-subspace of Q is a linear subspace \mathbb{P}(W) \subset Q for some subspace W \subseteq V with \phi|_W = 0, equivalently a totally isotropic subspace under b_\phi.^[37] The maximal such subspaces have dimension equal to the index of \phi, and over algebraically closed fields, nonsingular quadrics of dimension $2m-1 contain two families of m-1-dimensional q-subspaces.^[21] In general fields, the existence and dimension of rational q-subspaces depend on the anisotropy of \phi, with the Grassmannian of isotropic subspaces parametrizing lines or higher-dimensional flats on Q.^[37] A quadric Q is degenerate if \mathrm{rad} \, b_\phi \neq 0, in which case its projective variety is the union of the vertex \mathbb{P}(\mathrm{rad} \, b_\phi) and a lower-dimensional quadric on the quotient V / \mathrm{rad} \, b_\phi.^[37] For example, in \mathbb{P}^3_k, degenerate quadrics include pairs of planes (rank 2 form) or a double plane (rank 1), arising when the matrix of \phi has corank at least 1.^[38] More generally, the singular locus has dimension \dim \mathrm{rad} \, \phi - 1, and Q decomposes as a cone over a quadric of dimension one less than the corank allows.^[37]

Symmetries and Polar Spaces

The symmetries of a quadric in projective space are governed by the orthogonal group associated to the underlying quadratic form. For a non-degenerate quadratic form Q on a vector space V over a field F, the orthogonal group O(Q) consists of all linear automorphisms g \in \mathrm{GL}(V) such that Q(gv) = Q(v) for all v \in V.^[39] In the projective setting, the quadric is the hypersurface \{ \in \mathbb{P}(V) \mid Q(v) = 0 \}, and its automorphism group as a projective variety is the projective orthogonal group \mathrm{PGO}(Q) = O(Q) / Z(O(Q)), where Z(O(Q)) is the kernel of the action on \mathbb{P}(V), typically \{\pm I\} for characteristic not 2. These symmetries act as isometries preserving the geometric structure of the quadric. A representative example is the sphere in Euclidean three-space, which is a quadric defined by Q(x,y,z) = x^2 + y^2 + z^2 = 1. Its group of orientation-preserving isometries is the special orthogonal group \mathrm{SO}(3), a compact Lie group of dimension 3 that acts transitively on the sphere.^[39] More generally, over the reals, the full orthogonal group O(p,q) preserves indefinite quadratic forms of signature (p,q), yielding hyperbolic or elliptic geometries on the corresponding quadrics. The polar structure arises from the symmetric bilinear form B associated to Q, defined by B(u,v) = \frac{1}{2} (Q(u+v) - Q(u) - Q(v)) in characteristic not 2, which is non-degenerate if Q is.^[21] The polar map sends a point \in \mathbb{P}(V) to its polar hyperplane \{ \in \mathbb{P}(V) \mid B(x,y) = 0 \}, consisting of points perpendicular to x with respect to B. For a subspace U \subseteq V, the perpendicular space is U^\perp = \{ v \in V \mid B(u,v) = 0 \ \forall u \in U \}, satisfying U^{\perp\perp} = U and \dim U + \dim U^\perp = \dim V.^[21] This polarity interchanges points and hyperplanes, inducing a duality on the projective space. Polar spaces emerge as incidence geometries derived from these perpendicular structures on quadrics. In the classical construction, for a non-degenerate quadratic form on a vector space of dimension at least 3, the polar space has points as the maximal totally singular (isotropic) 1-dimensional subspaces (i.e., points on the quadric), and lines as the maximal totally singular 2-dimensional subspaces (lines lying on the quadric).^[40] Axioms ensure that any two points lie on at most one line, and for any point not on a given line, there is a unique line through it intersecting the given line.^[40] The orthogonal group O(Q) acts as the collineation group preserving this incidence structure. In higher dimensions, such polar spaces of rank r \geq 3 form the points and hyperplanes of spherical buildings associated to the orthogonal group, capturing the combinatorial geometry of maximal isotropic subspaces.^[41] This framework generalizes to quadratic sets, which are subsets S of a projective space such that S intersects every line in at most two points, and the induced structure satisfies polarity-like properties akin to quadrics. Quadrics serve as prototypical quadratic sets, embedding the polar space geometry into broader combinatorial structures over finite fields, where they yield finite geometries with applications in group theory and coding theory.

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