In mathematics, a quadric or quadric surface is a generalization of conic sections (ellipses, parabolas, and hyperbolas). In three-dimensional space, quadrics include ellipsoids, paraboloids, and hyperboloids.

More generally, a quadric hypersurface (of dimension D) embedded in a higher dimensional space (of dimension ) is defined as the zero set of an irreducible polynomial of degree two in variables; for example, D1 is the case of conic sections (plane curves). 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.

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.

Formulation

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, xT 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.

Euclidean plane

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

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:

which takes only the value 0 or 1.

Each of these 17 normal forms (imaginary ellipsoid (imaginary elliptic cylinder (pair of complex conjugate parallel planes, a reducible quadric). In one case, the imaginary cone 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 obtains a quadric of revolution, which remains invariant when rotated around an axis (or infinitely many axes, in the case of the sphere).

Intersection of a Ray with a Quadric Surface

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One can represent a three-dimensional ray parametrically. If there are two real roots, this gives the two intersection points of the ray with the quadric surface. If there is a double root, then the ray grazes the surface. If there are no real roots, then the ray misses the quadric surface.

Alternatively, using homogeneous coordinates, one may represent the ray asyielding the same quadratic equation in t.

Quadric surface patches in graphical ray tracing

In computer graphics, the visual representation of mathematically modeled objects is frequently determined by ray tracing. Frequently, the surfaces of the objects are described in terms of a number of points on the surface and the normals to the surface at those points. The intermediate points on the surface are described by a set of surface patches between the points. Quadric surfaces are frequently used this way when the surface is described as a set of triangles using the positions and normals to the points of those triangles. Using quadric surfaces to model these patches has the advantage that it is easy to tell if and where a light ray intersects a patch, as was described in the previous section.

linear matrix to designate these constraints. However, it can be shown that this matrix will always be singular, so some additional constraint must be added to uniquely determine the quadric surface of that patch.

Determining the quadric surface type and the displacement from standard position

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In general, quadric equations may not be in normal form, and it may be difficult to determine the type of quadric surface by visual inspection unless the quadric surface has its center at the origin and the principal axes are along the x, ythen the following table allows classification of the quadric surface, based on the ranks, determinants, and eigenvalues matrices., the others being zeroes. Here the x, y, and z are the normalized eigenvectors. where the acute accent implies the matrix transpose. The location of the centroidFor a real ellipsoid, radii for all of the eigenvalues are positive. For a hyperboloid of one sheet, one of the squared radii is negative, the other two being those of the ellipse at the neck of the hyperboloid. For a hyperboloid of two sheets, two of the squared radii are negative and the real radius is the distance from the center point to the points of the sheets. If all three eigenvalues give negative squared radii, then the quadric surface is an imaginary ellipsoid.

Quadric Surface Affine Transformation

Affine transformation matrices are used to represent coordinate transformations. Such a matrix has the form in the aircraft frame of reference is the transformation between the ground frame of reference and the aircraft frame of reference.

, is moved to is as viewed from, where the transformation is

, it does not matter whether or not this matrix is transposed, and one may see it either way in the literature.

Latitude, longitude, and altitude as quadric surfaces

Latitude and longitude are normally thought of as lines on the globe, but when you consider that one may cross a line of latitude or longitude at different altitudes, it becomes obvious that these lines are actually surfaces, in fact, quadric surfaces. If the Earth is approximated as a sphere, longitudes are planes slicing though the center of the Earth and through the poles, while latitudes are cones, with the point of the cone at the center of the Earth. Altitudes are spheres of different radii. This is the general model currently used for naval and aviation navigation.

When the Earth is better approximated as an ellipsoid, the longitudes are still the same planes as for the spherical Earth. Latitudes are still cones, but the points of the cones are no longer at the center of the Earth, except at the equator; the cone axis is still along the Earth's axis, but north or south of the Earth center. Altitudes are ellipsoids. In the World Geodetic System) as 6,356,752.3142 meters. The squared first eccentricity

, Earth-centered (geocentric) coordinates can be determined as where H is the height above the Earth ellipsoid

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 usual 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 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

. When the characteristic of the field 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 .

A quadric is said to be non-degenerate is invertible.

A non-degenerate quadric is non-singular in the sense that its projective completion has no singular point (a cylinder is non-singular in the affine space, but it is a degenerate quadric that has a singular point at infinity).

The singular points of a degenerate quadric are the points whose projective coordinates belong to the null space of the matrix .

A quadric is reducible if and only if the rank of is one (case of a double hyperplane) or two (case of two hyperplanes).

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.

Rational parametrization

Given a non-singular point of a quadric, a line passing through is either tangent to the quadric, or intersects the quadric in exactly one other point (as usual, a line contained in the quadric is considered as a tangent, since it is contained in the tangent hyperplane). This means that the lines passing through and not tangent to the quadric are in one to one correspondence with the points of the quadric that do not belong to the tangent hyperplane at . Expressing the points of the quadric in terms of the direction of the corresponding line provides parametric equations of the following forms.

In the case of conic sections (quadric curves), this parametrization establishes a bijection between a projective conic section and a projective line; this bijection is an isomorphism of algebraic curves. In higher dimensions, the parametrization defines a birational map, which is a bijection between dense open subsets of the quadric and a projective space of the same dimension (the topology that is considered is the usual one in the case of a real or complex quadric, or the Zariski topology in all cases). The points of the quadric that are not in the image of this bijection are the points of intersection of the quadric and its tangent hyperplane at .

In the affine case, the parametrization is a rational parametrization of the form

are polynomials of degree at most two.

In the projective case, the parametrization has the form

are homogeneous polynomials of degree two.

For computing the parametrization and proving that the degrees are as asserted, one may proceed as follows in the affine case. One can proceed similarly in the projective case.

Let be the coordinate vector be a vector defining the direction used for the parametrization (directions whose last coordinate is zero are not taken into account here; this means that some points of the affine quadric are not parametrized; one says often that they are parametrized by points at infinity such that

one obtains the desired parametrization as fractions of polynomials of degree at most two.

Example: circle and spheres

Let consider the quadric of equation

this is the unit circle this is the unit sphere; in higher dimensions, this is the unit hypersphere.

of the preceding section becomes

one obtains

and simplifying the expression of the last coordinate, one obtains the parametric equation

By homogenizing, one obtains the projective parametrization

and one has a bijection between the circle and the projective line.

In this specific case, these points have nonreal complex coordinates, but it suffices to change one sign in the equation of the quadric for producing real points that are not obtained with the resulting parametrization.

Rational points

A quadric is defined over a field of the rational numbers, one can suppose that the coefficients are integers by clearing denominators.

is said rational of the real numbers is called a real point.

is called simply a rational point. By clearing denominators, one can suppose and one supposes generally that the projective coordinates) are integers. Also, by clearing denominators of the coefficients, one supposes generally that all the coefficients of the equation of the quadric and the polynomials occurring in the parametrization are integers.

Finding the rational points of a projective quadric amounts thus to solving a Diophantine equation.

Given a rational point over a quadric over a field , the parametrization described in the preceding section provides rational points when the parameters are in , and, conversely, every rational point of the quadric can be obtained from parameters in , if the point is not in the tangent hyperplane at .

It follows that, if a quadric has a rational point, it has many other rational points (infinitely many if is infinite), and these points can be algorithmically generated as soon one knows one of them.

the parametrization takes the form

are homogeneous polynomials of degree two with integer coefficients. Because of the homogeneity, one can consider only parameters that are setwise coprime is the equation of the quadric, a solution of this equation is said primitive if its components are setwise coprime integers. The primitive solutions are in one to one correspondence with the rational points of the quadric (up to a change of sign of all components of the solution). The non-primitive integer solutions are obtained by multiplying primitive solutions by arbitrary integers; so they do not deserve a specific study. However, setwise coprime parameters can produce non-primitive solutions, and one may have to divide by a greatest common divisor to arrive at the associated primitive solution.

Pythagorean triples

This is well illustrated by Pythagorean triples. A Pythagorean triple is a triple A Pythagorean triple is primitive is coprime.

and , the above method provides the parametrization

(The names of variables and parameters are being changed from the above ones to those that are common when considering Pythagorean triples).

If and the resulting triple is a Pythagorean triple. If one of and is even and the other is odd, this resulting triple is primitive; otherwise, and are both odd, and one obtains a primitive triple by dividing by 2.

even are obtained as

with and (this is Euclid's formula odd are obtained as

with and

As the exchange of and transforms a Pythagorean triple into another Pythagorean triple, only one of the two cases is sufficient for producing all primitive Pythagorean triples up to the order of and .

Projective quadrics over fields

The definition of a projective quadric in a real projective space (see above) can be formally adapted by 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 by starting with a quadratic form on a vector space.

Quadratic form

be a field a vector space such that

1. (Q1).

   (Q2) is a bilinear form.

is called quadratic form is symmetric.

are mutually determined in a unique way. is symplectic.

has the familiar form

1. and

2. .

For example:

n-dimensional projective space over a field

,

1. an -dimensional vector space

2. the 1-dimensional \vec 0\ne \vec x\in V_{n+1},

3. the set of points ,

4. the set of lines.

5. is the -dimensional projective space.

6. is a -dimensional subspace. A 2-dimensional subspace is a plane.

7. -dimensional subspace is called hyperplane.

Projective quadric

defines a quadric. That is,

.: (E1): one obtains a conic. (E2):;

.

Polar space

the set

is called polar space).

.

is a non trivial linear equation which defines a hyperplane. Hence

1. is either a hyperplane.

Intersection with a line

, the following cases may occur:

1. is called exterior line

2. is called a line in the quadric

3. is called tangent line

4. is called secant line.

Proof: one obtains for any for any, which proves b) and b')., which proves c).

Additionally the proof shows:

1. is a tangent.

f-radical, q-radical

(see above) and so one has to deal with two radicals:

1. is called f-radical.

2. is called singular radical or -radical.

3. .

A quadric is called non-degenerate.

(see above): (E1):. In this case the f-radical is the common point of all tangents, the so called knot. and the quadric (conic) ist non-degenerate. (E2): the intersection point. In this example the quadric is degenerate.

Symmetries

A quadric is a rather homogeneous object:

1. there exists an involutorial central collineation.

Proof: is a hyperplane.

The linear mapping

induces an involutorial central collineation invariant. produces the familiar shape.

Remark:

1. a) An exterior line, a tangent line or a secant line on an exterior, tangent and secant line, respectively.

2. .

q-subspaces and index of a quadric

For example: points on a sphere or lines on a hyperboloid (s. below).

1. Any two maximal.

then

1. is called index.

Theorem:

1. the following is true:

   1. .

its index.

1. is called sphere (or oval).

2. is called hyperboloid (of one sheet).

Examples:

1. is non-degenerate with index 1.

2. is irreducible !).

3. 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 obtain secants bearing more than 2 points of the quadric which is totally different from usual quadrics. The reason is the following statement.

1. A division ring is commutative if and only if any equation, has at most two solutions.

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.

See also

• Klein quadric

• Rotation of axes

• Superquadrics

• Translation of axes

References

Bibliography

• M. Audin: Geometry, Springer, Berlin, 2002, , p.200.

• M. Berger: Problem Books in Mathematics, ISSN 0941-3502, Springer New York, pp. 79–84.

• A. Beutelspacher, U. Rosenbaum: Projektive Geometrie, Vieweg + Teubner, Braunschweig u. a. 1992, , p.159.

• P. Dembowski: Finite Geometries, Springer, 1968, , p.43.

External links

• Interactive Java 3D models of all quadric surfaces

• Lecture Note , p.117

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