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 (D + 1)-dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in D + 1 variables (D = 1 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.
which may be compactly written in vector and matrix notation as:
where x = (x1, x2, ..., xD+1) is a row vector, xT is the transpose of x (a column vector), Q is a (D + 1) × (D + 1) matrix and P is a (D + 1)-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.
In three-dimensional Euclidean space, quadrics have dimension D = 2, and are known as quadric surfaces. They are classified and named by their orbits under affine transformations. More precisely, 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 Euclidean transformation or a change of Cartesian coordinates allows putting the quadratic equation of the quadric into one of the following normal forms:
where the are either 1, –1 or 0, except 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: (imaginary ellipsoid), (imaginary elliptic cylinder), and (pair of complex conjugate parallel planes, a reducible quadric). In one case, the imaginary cone, there is a single point (). If one has a line (in fact two complex conjugate intersecting planes). For one has two intersecting planes (reducible quadric). For one has a double plane. For 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.
|Non-degenerate real quadric surfaces|
| Hyperboloid of one sheet
| Hyperboloid of two sheets
|Degenerate real quadric surfaces|
| Elliptic cone
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).
|Quadrics of revolution|
|Oblate and prolate spheroids (special cases of ellipsoid)|
|Sphere (special case of spheroid)|
|Circular paraboloid (special case of elliptic paraboloid)|
|Hyperboloid of revolution of one sheet (special case of hyperboloid of one sheet)|
|Hyperboloid of revolution of two sheets (special case of hyperboloid of two sheets)|
|Circular cone (special case of elliptic cone)|
|Circular cylinder (special case of elliptic cylinder)|
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.
is a polynomial of degree two that defines an affine quadric, then its projective completion is defined by homogenizing p into
(this is a polynomial, because the degree of p is two). The points of the projective completion are the points of the projective space whose projective coordinates are zeros of P.
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 X0 = 1:
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 will include points with , which are not also solutions of because these points in projective space correspond to points "at infinity" in affine space.
A quadric in an affine space of dimension n 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 p has the form
for a matrix with and running from 0 to . When the characteristic of the field of the coefficients is not two, generally is assumed; equivalently . When the characteristic of the field of the coefficients is two, generally is assumed when ; equivalently is upper triangular.
The equation may be shortened, as the matrix equation
The equation of the projective completion is almost identical:
The quadric is said to be non-degenerate if the matrix is invertible.
Normal form of projective quadrics
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 with a vector having rational components yields a vector with integer components that satisfies ; set where the multiplying factor is the smallest positive integer that clears all the denominators of the components of .
Furthermore, when the underlying matrix is invertible, any one solution to for with rational components can be used to find any other solution with rational components, as follows. Let for some values of and , both with integer components, and value . Writing for a non-singular symmetric matrix with integer components, we have that
then the two solutions to , when viewed as a quadratic equation in , will be , where the latter is non-zero and rational. In particular, if is a solution of and is the corresponding non-zero solution of then any for which (1) is not orthogonal to and (2) satisfies these three conditions and gives a non-zero rational value for .
In short, if one knows one solution with rational components then one can find many integer solutions where depends upon the choice of . Furthermore, the process is reversible! If both satisfies and satisfies then the choice of will necessarily produce . 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 
- (Q1) for any and .
- (Q2) is a bilinear form.
is called quadratic form. The bilinear form is symmetric.
In case of the bilinear form is , i.e. and are mutually determined in a unique way.
In case of (that means: ) the bilinear form has the property , i.e. is symplectic.
For and ( is a base of ) has the familiar form
n-dimensional projective space over a field
Let be a field, ,
- an (n + 1)-dimensional vector space over the field
- the 1-dimensional subspace generated by ,
- the set of points ,
- the set of lines.
- is the n-dimensional projective space over .
- The set of points contained in a -dimensional subspace of is a -dimensional subspace of . A 2-dimensional subspace is a plane.
- In case of a -dimensional subspace is called hyperplane.
For a quadratic form on a vector space a point is called singular if . The set
of singular points of is called quadric (with respect to the quadratic form ).
Examples in .:
(E1): For one gets a conic.
(E2): For one gets the pair of lines with the equations and , respectively. They intersect at point ;
For the considerations below it is assumed that .
For point the set
is called polar space of (with respect to ).
If for any , one gets .
If for at least one , the equation is a non trivial linear equation which defines a hyperplane. Hence
- is either a hyperplane or .
Intersection with a line
For the intersection of a line with a quadric the familiar statement is true:
- For an arbitrary line the following cases occur:
- a) and is called exterior line or
- b) and is called tangent line or
- b′) and is called tangent line or
- c) and is called secant line.
Let be a line, which intersects at point and is a second point on .
From one gets
I) In case of the equation holds and it is for any . Hence either for any or for any , which proves b) and b').
II) In case of one gets and the equation has exactly one solution . Hence: , which proves c).
Additionally the proof shows:
- A line through a point is a tangent line if and only if .
In the classical cases or there exists only one radical, because of and and are closely connected. In case of the quadric is not determined by (see above) and so one has to deal with two radicals:
- a) is a projective subspace. is called f-radical of quadric .
- b) is called singular radical or -radical of .
- c) In case of one has .
A quadric is called non-degenerate if .
Examples in (see above):
(E1): For (conic) the bilinear form is
In case of the polar spaces are never . Hence .
In case of the bilinear form is reduced to and . Hence In this case the f-radical is the common point of all tangents, the so called knot.
In both cases and the quadric (conic) ist non-degenerate.
(E2): For (pair of lines) the bilinear form is and the intersection point.
In this example the quadric is degenerate.
A quadric is a rather homogeneous object:
Proof: Due to the polar space is a hyperplane.
The linear mapping
induces an involutorial central collineation with axis and centre which leaves invariant.
In case of mapping gets the familiar shape with and for any .
- a) An exterior line, a tangent line or a secant line is mapped by the involution on an exterior, tangent and secant line, respectively.
- b) is pointwise fixed by .
q-subspaces and index of a quadric
A subspace of is called -subspace if
For example: points on a sphere or lines on a hyperboloid (s. below).
- Any two maximal -subspaces have the same dimension .
Let be the dimension of the maximal -subspaces of then
- The integer is called index of .
- For the index of a non-degenerate quadric in the following is true:
Let be a non-degenerate quadric in , and its index.
- In case of quadric is called sphere (or oval conic if ).
- In case of quadric is called hyperboloid (of one sheet).
- a) Quadric in with form is non-degenerate with index 1.
- b) If polynomial is irreducible over the quadratic form gives rise to a non-degenerate quadric in of index 1 (sphere). For example: is irreducible over (but not over !).
- c) In the quadratic form generates a hyperboloid.
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.
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