Quadric
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension "D") in a -dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in "D" + 1 variables ( in the case of conic sections). 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". In coordinates , the general quadric is thus defined by the algebraic  equation which may be compactly written in vector and matrix notation as: where is a row vector, "x" is the transpose of "x" (a column vector), "Q" is a matrix and "P" is a -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. 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 , below. Euclidean plane. As the dimension of  a Euclidean plane is two, quadrics in a Euclidean plane have dimension one and are thus plane curves. They are called "conic sections", or "conics". Euclidean space. In three-dimensional Euclidean space, quadrics have dimension two, and are known as quadric surfaces. Their quadratic equations have the form where formula_4 are real numbers, and at least one of , , and is nonzero. The quadric surfaces are classified and named by their shape, which corresponds to the orbits under affine transformations. That is, if an affine transformation maps a quadric onto another one, they belong to the same class, and share the  same name and many properties. The principal axis theorem shows that for any (possibly reducible) quadric, a suitable change of Cartesian coordinates or, equivalently, a Euclidean transformation allows putting the equation of the quadric into a unique simple form on which the class of the quadric is immediately visible. This form is called the normal form of the equation, since two quadrics have the same normal form if and only if there is a Euclidean transformation that maps one quadric to the other. The normal forms are as follows: where the formula_9 are either 1, –1 or 0, except formula_10  which takes only the value 0 or 1. Each of these 17 normal forms corresponds to a single orbit under affine transformations. In three cases there are no real points: formula_11 ("imaginary ellipsoid"), formula_12 ("imaginary elliptic cylinder"), and formula_13 (pair of complex conjugate parallel planes, a reducible quadric). In one case, the "imaginary cone", there is a single point (formula_14). If formula_15 one has a line (in fact two complex conjugate intersecting planes). For formula_16 one has two intersecting planes (reducible quadric). For formula_17 one has a double plane. For formula_18 one has two parallel planes (reducible quadric). Thus, among  the 17 normal forms, there are nine true quadrics: a cone, three cylinders (often called degenerate quadrics) and five non-degenerate quadrics (ellipsoid, paraboloids and hyperboloids), which are detailed in the following tables. The eight remaining quadrics are the imaginary ellipsoid (no real point), the imaginary cylinder (no real point), the imaginary cone (a single real point), and the reducible quadrics, which are decomposed in two planes; there are five such decomposed quadrics, depending whether the planes are distinct or not, parallel or not, real or complex conjugate. When two or more of the parameters of the canonical equation are  equal, one gets a quadric of revolution, which remains invariant when rotated around an axis (or infinitely many axes, in the case of the sphere). Definition and basic properties. An "affine quadric" is the set of zeros of a polynomial of degree two. When not specified otherwise, the polynomial is supposed to have real coefficients, and the zeros are points in a Euclidean space. However, most properties remain true when the coefficients belong to any field and the points belong in an affine space. As usually in algebraic geometry, it is often useful to consider points over an algebraically closed  field containing the polynomial coefficients, generally the complex numbers, when the coefficients are real. Many properties becomes easier to state (and to prove) by extending the quadric to the projective space by projective completion, consisting of adding points at infinity. Technically, if is a polynomial of degree two that defines an affine quadric, then its projective completion is defined by homogenizing into (this is a polynomial, because the degree of is two). The points of the projective completion are the points of the projective space whose projective coordinates are zeros of . So, a "projective quadric" is the set of zeros  in a projective space of a homogeneous polynomial of degree two. As the above process of homogenization can be reverted by setting : it is often useful to not distinguish an affine quadric from its projective completion, and to talk of the "affine equation" or the "projective equation" of a quadric. However, this is not a perfect equivalence; it is generally the case that formula_22 will include points with formula_23, which are not also solutions of formula_24 because these points in projective space correspond to points "at infinity" in affine space. Equation. A quadric in an affine space of dimension is  the set of zeros of a polynomial of degree 2. That is, it is the set of the points whose coordinates satisfy an equation where the polynomial has the form for a matrix formula_27 with formula_28 and formula_29 running from 0 to formula_30. When the characteristic of the field of the coefficients is not two, generally formula_31 is assumed; equivalently formula_32. When the characteristic of the field of the coefficients is two, generally formula_33 is assumed when formula_34; equivalently formula_35 is upper triangular. The equation may be shortened, as the matrix equation with The equation of the projective completion is  almost identical: with These equations define a quadric as an algebraic hypersurface of dimension and degree two in a space of dimension . The quadric is said to be non-degenerate if the matrix formula_35 is invertible. Normal form of projective quadrics. In real projective space, by Sylvester's law of inertia, a non-singular quadratic form "P"("X") may be put into the normal form by means of a suitable projective transformation (normal forms for singular quadrics can have zeros as well as ±1 as coefficients). For two-dimensional surfaces (dimension "D" = 2) in three-dimensional space, there are exactly  three non-degenerate cases: The first case is the empty set. The second case generates the ellipsoid, the elliptic paraboloid or the hyperboloid of two sheets, depending on whether the chosen plane at infinity cuts the quadric in the empty set, in a point, or in a nondegenerate conic respectively. These all have positive Gaussian curvature. The third case generates the hyperbolic paraboloid or the hyperboloid of one sheet, depending on whether the plane at infinity cuts it in two lines, or in a nondegenerate conic respectively. These are doubly ruled surfaces of negative Gaussian curvature. The degenerate form generates  the elliptic cylinder, the parabolic cylinder, the hyperbolic cylinder, or the cone, depending on whether the plane at infinity cuts it in a point, a line, two lines, or a nondegenerate conic respectively. These are singly ruled surfaces of zero Gaussian curvature. We see that projective transformations don't mix Gaussian curvatures of different sign. This is true for general surfaces. In complex projective space all of the nondegenerate quadrics become indistinguishable from each other. Integer and rational solutions. Each solution of formula_24 with formula_45 a vector having rational components yields a vector formula_46 with integer components that satisfies formula_22 ; set formula_48 where the multiplying factor formula_49 is the smallest positive integer that clears all the denominators of the components of formula_45. Furthermore, when the underlying matrix is invertible, any one solution to formula_51 for formula_52 with rational components can be used to find any other solution with rational components, as follows. Let formula_53 for some values of formula_54 and formula_55, both with integer components, and value formula_56. Writing formula_57 for a non-singular symmetric matrix formula_35 with integer components, we have that When then the two solutions to formula_22, when viewed as a quadratic equation in formula_56, will be  formula_63, where the latter is non-zero and rational. In particular, if formula_52 is a solution of formula_51 and formula_66 is the corresponding non-zero solution of formula_67 then any formula_68 for which (1) formula_68 is not orthogonal to formula_70 and (2) formula_71 satisfies these three conditions and gives a non-zero rational value for formula_56. In short, if one knows one solution formula_52 with rational components then one can find many integer solutions formula_74 where formula_75 depends upon the choice of formula_68. Furthermore, the process is reversible! If both formula_52 satisfies formula_51 and formula_79 satisfies formula_80 then the choice  of formula_81 will necessarily produce formula_82. With this approach one can generate all Pythagorean triples or Heronian triangles. Projective quadrics over fields. The definition of a projective quadric in a real projective space (see above) can be formally adopted defining a projective quadric in an n-dimensional projective space over a field. In order to omit dealing with coordinates a projective quadric is usually defined starting with a quadratic form on a vector space Quadratic form. Let formula_83 be a field and formula_84 a vector space over formula_83. A mapping formula_86 from formula_84 to formula_83 such that is called quadratic  form. The bilinear form formula_93 is symmetric"." In case of formula_94 the bilinear form is formula_95, i.e. formula_93 and formula_86 are mutually determined in a unique way. In case of formula_98 (that means: formula_99) the bilinear form has the property formula_100, i.e. formula_93 is "symplectic". For formula_102 and formula_103 (formula_104 is a base of formula_84) formula_106 has the familiar form For example: Section::::"n"-dimensional projective space over a field. Let formula_83 be a field, formula_111, Projective quadric. For a quadratic form formula_86 on a vector space formula_112 a point formula_128 is called "singular" if formula_129. The set of singular points of  formula_86 is called quadric (with respect to the quadratic form formula_86). Examples in formula_133.: (E1): For formula_134 one gets a conic. (E2): For formula_135 one gets the pair of lines with the equations formula_136 and formula_137, respectively. They intersect at point formula_138; For the considerations below it is assumed that formula_139. Polar space. For point formula_140 the set is called polar space of formula_142 (with respect to formula_86). If formula_144 for any formula_145, one gets formula_146. If formula_147 for at least one formula_145, the equation formula_144is a non trivial linear equation which defines a hyperplane. Hence Intersection with a line . For the intersection of a line with a quadric formula_152 the familiar statement is true: Proof: Let formula_153 be a line, which intersects formula_163 at point formula_164 and formula_165 is a second point on formula_153. From formula_167 one gets formula_168 I) In case of formula_169 the equation formula_170 holds and it is formula_171 for any formula_172. Hence either formula_173 for "any" formula_172 or formula_175 for "any" formula_172, which proves b) and b'). II) In case of formula_177 one gets formula_178 and the equation formula_179 has exactly one solution formula_180. Hence: formula_181, which proves c). Additionally the proof shows: Section::::"f"-radical, "q"-radical . In the classical cases formula_185 or formula_186 there exists only one radical, because of formula_94 and formula_93 and formula_86 are closely connected. In case of formula_98 the quadric formula_191 is not determined by formula_93 (see above) and so one has to deal with two radicals: A quadric is called non-degenerate if formula_201. Examples in formula_133 (see above): (E1): For formula_134 (conic) the bilinear form is formula_204 In case of formula_94 the polar spaces are never formula_206. Hence formula_207. In case of formula_98 the bilinear form is reduced to formula_209 and formula_210. Hence formula_211 In this case the "f"-radical is  the common point of all tangents, the so called "knot". In both cases formula_212 and the quadric (conic) ist "non-degenerate". (E2): For formula_135 (pair of lines) the bilinear form is formula_214 and formula_215 the intersection point. In this example the quadric is "degenerate". Symmetries. A quadric is a rather homogeneous object: Proof: Due to formula_220 the polar space formula_150 is a hyperplane. The linear mapping induces an "involutorial central collineation" formula_217 with axis formula_150 and centre formula_142 which leaves formula_191 invariant. In case of formula_94 mapping formula_228 gets the familiar shape formula_229 with formula_230 and formula_231 for any formula_232 . Remark: Section::::"q"-subspaces and index of a quadric. A subspace formula_236 of formula_123 is called formula_86-subspace if formula_239 For example: points on a sphere or lines on a hyperboloid (s. below). Let be formula_241 the dimension of the maximal formula_86-subspaces of formula_191 then Theorem: (BUEKENHOUT) Let be formula_191 a non-degenerate quadric in formula_252, and formula_28 its index. Examples: Generalization of quadrics: quadratic sets. It is not reasonable to formally extend the definition of quadrics to spaces over genuine skew fields (division rings). Because one would get secants bearing more than 2 points of the quadric which is totally  different from "usual" quadrics. The reason is the following statement. There are "generalizations" of quadrics: quadratic sets. A quadratic set is a set of points of a projective space with the same geometric properties as a quadric: every line intersects a quadratic set in at most two points or is contained in the set.