Quadric

A quadric (from the Latin square square) is in mathematics the solution set of a quadratic equation of multiple unknown. In two dimensions, a quadric generally forms a curve in the plane, which is then to be a conic section. In three dimensions, describes a quadric generally an area in space, which is also known as second-order or quadratic surface area. Generally, these are at a quadric to an algebraic variety, ie a special hypersurface in a finite real coordinate space. Through a principal axis transformation can be any quadric transform on one of three possible normal forms. In this way quadrics can be classified into several basic types.

Quadrics are investigated in particular in the analytic and projective geometry. Applications for quadrics in science and technology can be found among others in the geodesy ( reference ellipsoid ), the architecture (structural design ), or optics ( parabolic ).

• 4.1 quadrics in one dimension
• 4.2 quadrics in the plane
• 4.3 quadrics in space

Definition

A quadric is a set of points in the - dimensional real coordinate space of the form

In which

A quadratic polynomial in the variables is. At least one of the polynomial has to be non-zero. In addition, it can be assumed without restriction that applies to everyone. A quadric is thus the set of zeros of a quadratic polynomial of several variables or the set of solutions of a quadratic equation with several unknowns.

Examples

For example, describes the set of points

An ellipse in the plane. The set of points

Describes a single-shell hyperboloid in three dimensions.

Properties

Matrix representation

In compact matrix notation, a quadric as a set of vectors

Be described with a symmetric matrix and and are column vectors of appropriate length. Using the advanced representation matrix

And the corresponding extended vector can be a quadric also compact by the amount

Be represented in homogeneous coordinates.

Types

In quadrics three basic types are distinguished. The decision as to what type it is at a given quadric may be inferred from the ranks of the matrices, and can be made:

• Tapered Type:
• Mittelpunktsquadrik:
• Parabolic type:

A quadric is called degenerate case, if

Applies. While nondegenerate quadrics form curved hypersurfaces in all directions, point degenerate quadrics in some directions rectilinear structures on or otherwise degenerated.

Transformations

Quadrics can be transformed by similarity images without changing their type result. Is a regular matrix, then is obtained by the linear transformation a new quadric in the coordinates of equation

Sufficient. Likewise, obtained by a parallel translation by a vector a new quadric, the equation

Filled with the identity matrix. In particular, the rank of the matrices and by such affinities do not change.

Normal Forms

Through a principal axis transformation can be any quadric transform one of the following normal forms. For this purpose, first, an orthogonal matrix, for example a rotating or mirroring matrix selected so that the diagonal matrix which contains the eigenvalues ​​in descending order. In the second step, the transformed quadric is shifted by a vector such that the linear terms and the constant term largely disappear. Finally, the quadric is still normalized so that the constant term, if it is not null, is to one. This results in the following three normal forms:

• Tapered Type: with
• Mittelpunktsquadrik: with
• Parabolic type: with

In addition as a special case the

• Empty set: with

In all cases, the coefficients. The ratios and result in this case from the signature of the matrix.

Classification

Quadrics in one dimension

In one dimension, a quadric is the set of solutions of a quadratic equation in one unknown, which is a point set of the form

By moving ( completing the square ) and normalization to distinguish between the following two cases:

In the remaining case, it appears as a solution set is the empty set. In all cases.

Quadrics in the plane

In the plane, a quadric is the set of solutions of a quadratic equation with two unknowns, which is a point set of the form

This is except for degenerate cases to conic sections, with degenerate conic sections, in which the cone tip is included in the section plane can be distinguished from non-degenerate conic sections. By principal axis transformation can be the general equation of a quadric on one of the following normal forms transform:

In the remaining two cases, and is calculated as the amount of solution in each case the empty set. In all cases.

Quadrics in space

In three-dimensional space a quadric is the set of solutions of a quadratic equation with three unknowns, which is a point set of the form

In space the variety of quadrics is significantly larger than in the plane. There are also degenerate and is not arte tete quadrics. Among the degenerate quadrics thereby find simple curved surfaces, such as cylinders and cones. Just as in two dimensions can be the general equation of a quadric on one of the following normal forms transform:

In the three remaining cases, and is calculated as the amount of solution in each case in turn the empty set. In all cases.

For (or in the case of the two -sheeted hyperboloid ) is obtained in the following cases, surfaces of revolution, which are also known as rotary quadrics: ellipsoid, one-and two-shell hyperboloid of revolution, of revolution, circular cone and a circular cylinder. Ruled surfaces, ie surfaces that are generated by a one-parameter family of straight lines, are cone, elliptic and parabolic cylinders, planes, monocoque and hyperbolic paraboloid hyperboloid. The latter three areas are produced even by two straight lines and crowds are the only possible rule double curved surfaces in space.

Projective quadrics

The variety of quadrics is significantly reduced when one completes both the affine space in which a quadric is defined, as well as the quadric itself projective. The projective extensions of ellipses, hyperbolas and parabolas are all projectively equivalent to each other, ie there is a projective collineation which maps one curve to the other (see projective conic ).

In three-dimensional space are the following quadrics are equivalent:

• Ellipsoid, two-shell hyperboloid and elliptic paraboloid,
• Single-shell hyperboloid, and hyperbolic paraboloid,
• Elliptic, hyperbolic, parabolic cylinders and cones.

Generalizations

General can be considered quadrics in vector spaces over an arbitrary field, including over the field of complex numbers or over finite fields.