Quadric

Quadrics in three-dimensional space: single- and double-shell hyperboloid , ellipsoid , hyperbolic paraboloid , cylinder , elliptical paraboloid and cone (from left to right)

In mathematics, a quadric (from the Latin quadra square) is the set of solutions to a quadratic equation of several unknowns. In two dimensions, a quadric usually forms a curve in the plane , which is then a conic section . In three dimensions, a quadric generally describes an area in space , which is also called a second-order area or a square area . In general, a quadric is an algebraic variety , i.e. a special hypersurface in a finite-dimensional real coordinate space . Each quadric can be transformed to one of three possible normal forms by means of a major axis transformation. In this way, quadrics can be classified into several basic types.

Quadrics are particularly studied in analytical and projective geometry . Applications for quadrics in technology and the natural sciences can be found in geodesy ( reference ellipsoid ), architecture ( supporting structure ) or optics ( parabolic mirrors ).

definition

A quadric is a set of points in the -dimensional real coordinate space of the form ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle Q=\left\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}\mid q(x_{1},\ldots ,x_{n})=0\right\}}$,

in which

${\displaystyle q(x_{1},\ldots ,x_{n})=\sum _{i,j=1}^{n}a_{ij}x_{i}x_{j}+2\,\sum _{i=1}^{n}b_{i}x_{i}+c}$

is a quadratic polynomial in variables . At least one of the polynomial coefficients must be non-zero. In addition, it can be assumed without restriction that applies to all . A quadric is the set of zeros of a quadratic polynomial of several variables or the set of solutions of a quadratic equation with several unknowns. ${\displaystyle x_{1},\ldots ,x_{n}}$ ${\displaystyle a_{11},\dots ,a_{nn}}$ ${\displaystyle a_{ij}=a_{ji}}$ ${\displaystyle i,j\in \{1,\dotsc ,n\}}$

Examples

For example, describes the set of points

${\displaystyle Q=\left\{(x,y)\in \mathbb {R} ^{2}\mid 2x^{2}+3y^{2}=5\right\}}$

an ellipse in the plane. The amount of points

${\displaystyle Q=\left\{(x,y,z)\in \mathbb {R} ^{3}\mid x^{2}+y^{2}-z^{2}=1\right\}}$

describes a single-shell hyperboloid in three-dimensional space.

properties

Matrix display

In compact matrix notation , a quadric can be used as a set of vectors

${\displaystyle Q=\left\{x\in \mathbb {R} ^{n}\mid x^{T}Ax+2b^{T}x+c=0\right\}}$

are described, where a symmetrical matrix and and column vectors are of corresponding length. With the help of the extended display matrix ${\displaystyle A=(a_{ij})\in \mathbb {R} ^{n\times n}}$ ${\displaystyle b=(b_{i})\in \mathbb {R} ^{n}}$ ${\displaystyle x=(x_{i})\in \mathbb {R} ^{n}}$

${\displaystyle {\bar {A}}={\begin{pmatrix}A&b\\b^{T}&c\end{pmatrix}}}$

and a correspondingly expanded vector can also make a quadric compact by the set ${\displaystyle {\bar {x}}={\tbinom {x}{1}}}$

${\displaystyle Q=\left\{x\in \mathbb {R} ^{n}\mid {\bar {x}}^{T}{\bar {A}}\,{\bar {x}}=0\right\}}$

are represented in homogeneous coordinates .

Types

There are three basic types of quadrics. The decision of what type a given quadric is can be made based on the ranks of the matrices , and : ${\displaystyle A}$ ${\displaystyle (A\mid b)}$ ${\displaystyle {\bar {A}}}$

• Conical type : ${\displaystyle \operatorname {rang} ({\bar {A}})=\operatorname {rang} (A\mid b)=\operatorname {rang} (A)}$
• Center square : ${\displaystyle \operatorname {rang} ({\bar {A}})>\operatorname {rang} (A\mid b)=\operatorname {rang} (A)}$
• Parabolic type : ${\displaystyle \operatorname {rang} (A\mid b)>\operatorname {rang} (A)}$

A quadric is called degenerate , if

${\displaystyle \det {\bar {A}}=0}$

applies. While non-degenerate quadrics form curved hypersurfaces in all directions, degenerate quadrics have linear structures in some directions or are otherwise degenerate.

Transformations

Quadrics can be transformed through similarity maps without changing their type. If it is a regular matrix , the linear transformation gives a new quadric in the coordinates , that of the equation ${\displaystyle S\in \mathbb {R} ^{n\times n}}$ ${\displaystyle y=S^{-1}x}$ ${\displaystyle y_{1},\ldots ,y_{n}}$

${\displaystyle {\begin{pmatrix}y^{T}\!\!&1\end{pmatrix}}{\begin{pmatrix}S^{T}&0\\0&1\end{pmatrix}}{\begin{pmatrix}A&b\\b^{T}&c\end{pmatrix}}{\begin{pmatrix}S&0\\0&1\end{pmatrix}}{\begin{pmatrix}y\\1\end{pmatrix}}={\begin{pmatrix}y^{T}\!\!&1\end{pmatrix}}{\begin{pmatrix}S^{T}AS&S^{T}b\\b^{T}S&c\end{pmatrix}}{\begin{pmatrix}y\\1\end{pmatrix}}=0}$

enough. A parallel shift by a vector also gives a new quadric, which is the equation ${\displaystyle y=x-u}$ ${\displaystyle u\in \mathbb {R} ^{n}}$

${\displaystyle {\begin{pmatrix}y^{T}\!\!&1\end{pmatrix}}{\begin{pmatrix}I&0\\u^{T}&1\end{pmatrix}}{\begin{pmatrix}A&b\\b^{T}&c\end{pmatrix}}{\begin{pmatrix}I&u\\0&1\end{pmatrix}}{\begin{pmatrix}y\\1\end{pmatrix}}={\begin{pmatrix}y^{T}\!\!&1\end{pmatrix}}{\begin{pmatrix}A&Au+b\\u^{T}A+b^{T}&u^{T}Au+2b^{T}u+c\end{pmatrix}}{\begin{pmatrix}y\\1\end{pmatrix}}=0}$

met with the identity matrix . In particular, the rank of the matrices and such affinities do not change. ${\displaystyle I\in \mathbb {R} ^{n\times n}}$ ${\displaystyle A,(A\mid b)}$ ${\displaystyle {\bar {A}}}$

If so, both methods can be combined using and : ${\displaystyle \det(A)\neq 0}$ ${\displaystyle y=S^{-1}x}$ ${\displaystyle z=y+S^{-1}A^{-1}b}$ ${\displaystyle z=S^{-1}(x+A^{-1}b)}$

${\displaystyle {\begin{pmatrix}z^{T}\!\!&1\end{pmatrix}}{\begin{pmatrix}S^{T}&0\\-b^{T}A^{-1}&1\end{pmatrix}}{\begin{pmatrix}A&b\\b^{T}&c\end{pmatrix}}{\begin{pmatrix}S&-A^{-1}b\\0&1\end{pmatrix}}{\begin{pmatrix}z\\1\end{pmatrix}}={\begin{pmatrix}z^{T}\!\!&1\end{pmatrix}}{\begin{pmatrix}S^{T}AS&0\\0&-b^{T}A^{-1}b+c\end{pmatrix}}{\begin{pmatrix}z\\1\end{pmatrix}}=0.}$

Since the matrix is symmetric, it is orthogonally diagonalizable, that is, there is an orthogonal matrix such that is a diagonal matrix. With this the quadric can through the condition ${\displaystyle A}$ ${\displaystyle S}$ ${\displaystyle S^{-1}AS=S^{T}AS=:D}$

${\displaystyle z^{T}Dz-b^{T}A^{-1}b+c=0}$

be expressed. So there are no longer any mixed-quadratic or linear terms. The center of the quadric is thus included . ${\displaystyle z=0\Leftrightarrow x=-A^{-1}b}$

Normal forms

Each quadric can be transformed to one of the following normal forms using a major axis transformation. For this purpose, an orthogonal matrix , for example a rotation or reflection matrix , is first selected in such a way that a diagonal matrix results which contains the eigenvalues of in descending order. In the second step, the transformed quadric is shifted by a vector in such a way that the linear terms and the constant term also largely disappear. Finally, the quadric is normalized in such a way that the constant term, if it is not zero, becomes one. This results in the following three normal forms: ${\displaystyle S}$ ${\displaystyle S^{T}AS}$ ${\displaystyle A}$ ${\displaystyle u}$

• Conical type:   with   ${\displaystyle {\frac {x_{1}^{2}}{\alpha _{1}^{2}}}+\dotsb +{\frac {x_{p}^{2}}{\alpha _{p}^{2}}}-{\frac {x_{p+1}^{2}}{\alpha _{p+1}^{2}}}-\dotsb -{\frac {x_{r}^{2}}{\alpha _{r}^{2}}}=0}$ ${\displaystyle 1\leq p\leq r\leq n,\ p\geq r-p}$
• Center square:   with   ${\displaystyle {\frac {x_{1}^{2}}{\alpha _{1}^{2}}}+\dotsb +{\frac {x_{p}^{2}}{\alpha _{p}^{2}}}-{\frac {x_{p+1}^{2}}{\alpha _{p+1}^{2}}}-\dotsb -{\frac {x_{r}^{2}}{\alpha _{r}^{2}}}=1}$ ${\displaystyle 1\leq p\leq r\leq n}$
• Parabolic type:   with   ${\displaystyle {\frac {x_{1}^{2}}{\alpha _{1}^{2}}}+\dotsb +{\frac {x_{p}^{2}}{\alpha _{p}^{2}}}-{\frac {x_{p+1}^{2}}{\alpha _{p+1}^{2}}}-\dotsb -{\frac {x_{r}^{2}}{\alpha _{r}^{2}}}-2x_{r+1}=0}$ ${\displaystyle 1\leq p\leq r<n,\ p\geq r-p}$

In addition, there is the special case

• Empty set:   with   ${\displaystyle -{\frac {x_{1}^{2}}{\alpha _{1}^{2}}}-\dotsb -{\frac {x_{r}^{2}}{\alpha _{r}^{2}}}=1}$ ${\displaystyle 1\leq r\leq n}$

In all cases the coefficients are . The key figures and result from the signature of the matrix . ${\displaystyle \alpha _{1},\dotsc ,\alpha _{r}>0}$ ${\displaystyle p=|\{\lambda \in \sigma (A)\colon \lambda >0\}|}$ ${\displaystyle r=|\{\lambda \in \sigma (A)\colon \lambda \neq 0\}|=\operatorname {rang} (A)}$ ${\displaystyle A}$

classification

Quadrics in one dimension

In one dimension, a quadric is the set of solutions to a quadratic equation with an unknown, i.e. a set of points of the form

${\displaystyle Q=\left\{x\in \mathbb {R} \mid ax^{2}+bx+c=0\right\}}$.

The following two cases can be distinguished by shifting ( square addition ) and normalization:

Non-degenerate quadrics		Degenerate quadrics
Two solutions ${\displaystyle {x^{2} \over \alpha ^{2}}=1}$		A solution ${\displaystyle {x^{2} \over \alpha ^{2}}=0}$

In the remaining case , the solution set is the empty set. In all cases it is . ${\displaystyle -{\tfrac {x^{2}}{\alpha ^{2}}}=1}$ ${\displaystyle \alpha >0}$

Quadrics in the plane

In the plane, a quadric is the set of solutions to a quadratic equation with two unknowns, i.e. a set of points of the form

${\displaystyle Q=\left\{(x,y)\in \mathbb {R} ^{2}\mid ax^{2}+bxy+cy^{2}+dx+ey+f=0\right\}}$.

Except for degenerate cases, these are conic sections , with degenerate conic sections, in which the cone tip is contained in the cutting plane, and non-degenerate conic sections are distinguished. The general equation of a quadric can be transformed to one of the following normal forms using principal axis transformation:

Non-degenerate quadrics		Degenerate quadrics
ellipse ${\displaystyle {x^{2} \over \alpha ^{2}}+{y^{2} \over \beta ^{2}}=1}$		Two intersecting straight lines ${\displaystyle {x^{2} \over \alpha ^{2}}-{y^{2} \over \beta ^{2}}=0}$
hyperbole ${\displaystyle {x^{2} \over \alpha ^{2}}-{y^{2} \over \beta ^{2}}=1}$		Two parallel straight lines ${\displaystyle {x^{2} \over \alpha ^{2}}=1}$
parabola ${\displaystyle {x^{2} \over \alpha ^{2}}-2y=0}$		A straight ${\displaystyle {x^{2} \over \alpha ^{2}}=0}$
		One point ${\displaystyle {\frac {x^{2}}{\alpha ^{2}}}+{\frac {y^{2}}{\beta ^{2}}}=0}$

In the two remaining cases and , the solution set is the empty set. In all cases are . ${\displaystyle -{\tfrac {x^{2}}{\alpha ^{2}}}-{\tfrac {y^{2}}{\beta ^{2}}}=1}$ ${\displaystyle -{\tfrac {x^{2}}{\alpha ^{2}}}=1}$ ${\displaystyle \alpha ,\beta >0}$

Quadrics in space

In three-dimensional space, a quadric is the solution set of a quadratic equation with three unknowns, i.e. a point set of the form

${\displaystyle Q=\left\{(x,y,z)\in \mathbb {R} ^{3}\mid ax^{2}+bxy+cxz+dy^{2}+eyz+fz^{2}+gx+hy+iz+j=0\right\}}$.

In space, the variety of quadrics is significantly greater than in the plane. There are degenerate and non-degenerate quadrics here as well. The degenerate quadrics also include simply curved surfaces such as cylinders and cones. Similar to two dimensions, the general equation of a quadric can be transformed to one of the following normal forms:

Non-degenerate quadrics		Degenerate quadrics (curved surfaces)		Degenerate quadrics (levels, etc.)
Ellipsoid ${\displaystyle {x^{2} \over \alpha ^{2}}+{y^{2} \over \beta ^{2}}+{z^{2} \over \gamma ^{2}}=1}$		Elliptical cone ${\displaystyle {x^{2} \over \alpha ^{2}}+{y^{2} \over \beta ^{2}}-{z^{2} \over \gamma ^{2}}=0}$		Two intersecting planes ${\displaystyle {\frac {x^{2}}{\alpha ^{2}}}-{\frac {y^{2}}{\beta ^{2}}}=0}$
Single-shell hyperboloid ${\displaystyle {x^{2} \over \alpha ^{2}}+{y^{2} \over \beta ^{2}}-{z^{2} \over \gamma ^{2}}=1}$		Elliptical cylinder ${\displaystyle {x^{2} \over \alpha ^{2}}+{y^{2} \over \beta ^{2}}=1}$		Two parallel planes ${\displaystyle {\frac {x^{2}}{\alpha ^{2}}}=1}$
Double-shell hyperboloid ${\displaystyle {x^{2} \over \alpha ^{2}}+{y^{2} \over \beta ^{2}}-{z^{2} \over \gamma ^{2}}=-1}$		Hyperbolic cylinder ${\displaystyle {x^{2} \over \alpha ^{2}}-{y^{2} \over \beta ^{2}}=1}$		A level ${\displaystyle {x^{2} \over \alpha ^{2}}=0}$
Elliptical paraboloid ${\displaystyle {x^{2} \over \alpha ^{2}}+{y^{2} \over \beta ^{2}}-2z=0}$		Parabolic cylinder ${\displaystyle {x^{2} \over \alpha ^{2}}-2y=0}$		A straight ${\displaystyle {\frac {x^{2}}{\alpha ^{2}}}+{\frac {y^{2}}{\beta ^{2}}}=0}$
Hyperbolic paraboloid ${\displaystyle {x^{2} \over \alpha ^{2}}-{y^{2} \over \beta ^{2}}-2z=0}$				One point ${\displaystyle {\frac {x^{2}}{\alpha ^{2}}}+{\frac {y^{2}}{\beta ^{2}}}+{\frac {z^{2}}{\gamma ^{2}}}=0}$

In the three remaining cases , and the empty set is the result of the solution set. In all cases are . ${\displaystyle -{\tfrac {x^{2}}{\alpha ^{2}}}-{\tfrac {y^{2}}{\beta ^{2}}}-{\tfrac {z^{2}}{\gamma ^{2}}}=1}$ ${\displaystyle -{\tfrac {x^{2}}{\alpha ^{2}}}-{\tfrac {y^{2}}{\beta ^{2}}}=1}$ ${\displaystyle -{\tfrac {x^{2}}{\alpha ^{2}}}=1}$ ${\displaystyle \alpha ,\beta ,\gamma >0}$

For (or is obtained in the case of the two-sheeted hyperboloid) in the following cases, surfaces of revolution , also called rotational quadric be referred to: ellipsoid , single- and double-shelled hyperboloid of revolution , paraboloid of revolution , a circular cone , and a circular cylinder . Ruled surfaces , i.e. surfaces that are generated by a single -parameter family of straight lines , are cones, elliptical and parabolic cylinders, planes, single-shell hyperboloid and hyperbolic paraboloid. The latter three surfaces are even generated by two sets of straight lines and are the only possible double-curved ruled surfaces in space. ${\displaystyle \alpha =\beta }$ ${\displaystyle \beta =\gamma }$

Projective Quadrics

→ Main article : Projective quadric

The diversity of the quadrics is considerably reduced if both the affine space in which a quadric is defined and the quadric itself are projectively closed . The projective extensions of ellipses, hyperbolas and parabolas are all projectively equivalent to one another, that is, there is a projective collineation that maps one curve onto the other (see projective conic section ).

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

• Ellipsoid, double-shell hyperboloid and elliptical paraboloid,
• single-shell hyperboloid and hyperbolic paraboloid,
• elliptical, hyperbolic, parabolic cylinder and cone.

Generalizations

More generally, quadrics can also be viewed in vector spaces over any body , i.e. also over the body of complex numbers or even over finite bodies .

Individual evidence

1. ^ ^{A b} Tilo Arens, Frank Hettlich, Christian Karpfinger, Ulrich Kockelkorn, Klaus Lichtenegger, Hellmuth Stachel : Mathematics . 2nd Edition. Spektrum Akademischer Verlag, 2011, ISBN 3-8274-2347-3 , pp. 719 .
2. ^ Kurt Meyberg, Peter Vachenauer: Higher Mathematics 1 . 6th edition. Springer, 2003, ISBN 978-3-540-41850-4 , pp. 345 .
3. ↑ Hanfried Lenz : Lectures on projective geometry. Academic publishing company Geest & Portig, Leipzig 1965, p. 155.

literature

• Ilja Nikolajewitsch Bronstein, Konstantin A. Semendjajew: Pocket book of mathematics. Teubner-Verlag, Leipzig 1983, ISBN 3-87144-492-8 , p. 283.
• Klemens Burg, Herbert Haf, Friedrich Wille: Higher Mathematics for Engineers. Volume II, Teubner-Verlag, Stuttgart, ISBN 3-519-22956-0 , p. 341.
• dtv atlas on mathematics. Volume 1, Deutscher Taschenbuch-Verlag, ISBN 3-423-03007-0 , pp. 200-203.
• Kurt Meyberg, Peter Vachenauer: Höhere Mathematik 1. Springer-Verlag, Berlin 1995, ISBN 3-540-59188-5 , p. 343.

Web links

Commons : Quadric surfaces  - collection of pictures, videos and audio files

Wiktionary: Quadrik  - explanations of meanings, word origins, synonyms, translations

• VS Malakhovskii: Quadric . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
• Eric W. Weisstein : Quadratic Surface . In: MathWorld (English).
• pahio: Quadratic Surfaces . In: PlanetMath . (English)
• Images of quadrics in space
• Interactive 3D models of all quadrics