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

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 = xu}$${\ 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}$${\ displaystyle 1 \ leq p \ leq r \ leq n, \ p \ geq rp}$
• Center square:   with   ${\ displaystyle {\ frac {x_ {1} ^ {2}} {\ alpha _ {1} ^ {2}}} + \ dotsb + {\ frac {x_ {p} ^ {2}} {\ alpha _ {p} ^ {2}}} - {\ frac {x_ {p + 1} ^ {2}} {\ alpha _ {p + 1} ^ {2}}} - \ dotsb - {\ frac {x_ { r} ^ {2}} {\ alpha _ {r} ^ {2}}} = 1}$${\ 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

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 {rank} (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:

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:

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}$

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.