Affine figure

In geometry and in linear algebra , sub-areas of mathematics , an affine mapping (also called affine transformation , especially in the case of a bijective mapping) is a mapping between two affine spaces in which collinearity , parallelism and partial relationships are preserved or become obsolete. More precisely formulated:

1. The images of points that lie on a straight line (ie are collinear) are again on a straight line ( invariance of collinearity ). All - but then all and not just some - points of a straight line can be mapped onto one point.
2. The images of two parallel straight lines are parallel if neither of the two straight lines is mapped onto a point.
3. Three different points that lie on a straight line (collinear points) are mapped in such a way that the partial ratio of their image points corresponds to that of the original image points - unless all three are mapped onto the same image point.

A bijective affine mapping of an affine space to itself is called affinity .

In school mathematics and some areas of application (for example in statistics, see below), special affine maps are also called linear mapping or linear functions . In general mathematical parlance, however, a linear mapping is a homomorphism of vector spaces .

definition

A mapping between affine spaces and is called affine mapping if there is a linear mapping between the associated vector spaces such that ${\displaystyle f\colon A\to B}$ ${\displaystyle (A,V_{A})}$ ${\displaystyle (B,V_{B})}$ ${\displaystyle \varphi \colon V_{A}\to V_{B}}$

${\displaystyle {\overrightarrow {f(P)f(Q)}}=\varphi \left({\overrightarrow {PQ}}\right)}$

applies to all points . And denote the connection vectors of the original image or image points. ${\displaystyle P,Q\in A}$ ${\displaystyle {\overrightarrow {PQ}}\in V_{A}}$ ${\displaystyle {\overrightarrow {f(P)f(Q)}}\in V_{B}}$

In the important application that and applies, a mapping is already an affine mapping if there is a linear mapping with ${\displaystyle A=V_{A}}$ ${\displaystyle B=V_{B}}$ ${\displaystyle f\colon A\to B}$ ${\displaystyle \varphi \colon V_{A}\to V_{B}}$

${\displaystyle f(P)=f(0)+\varphi (P)}$

for everyone in . In this case, an affine mapping is created by a translation of a linear mapping with the image of the zero point. ${\displaystyle P}$ ${\displaystyle A}$ ${\displaystyle f(0)}$

properties

• The linear mapping from the definition is clearly determined by. It is referred to below as . ${\displaystyle \varphi }$ ${\displaystyle f}$ ${\displaystyle \varphi _{f}}$
• A map is affine if and only if there is one such that the map ${\displaystyle f\colon A\to B}$ ${\displaystyle P_{0}\in A}$

${\displaystyle \varphi _{f}\colon V_{A}\to V_{B},\quad {\overrightarrow {P_{0}Q}}\mapsto {\overrightarrow {f(P_{0})f(Q)}}}$

is linear.

• If and as well as a linear mapping are given, there is exactly one affine mapping with and . ${\displaystyle P_{0}\in A}$ ${\displaystyle Q_{0}\in B}$ ${\displaystyle \psi \colon V_{A}\to V_{B}}$ ${\displaystyle f\colon A\to B}$ ${\displaystyle f(P_{0})=Q_{0}}$ ${\displaystyle \varphi _{f}=\psi }$
• An affine mapping is bijective if and only if is bijective. In this case the inverse mapping is also affine and it holds . ${\displaystyle f}$ ${\displaystyle \varphi _{f}}$ ${\displaystyle f^{-1}\colon B\to A}$ ${\displaystyle \varphi _{f^{-1}}=(\varphi _{f})^{-1}}$
• If there is another affine space and are , affine, then is also affine and it holds . ${\displaystyle (C,V_{C})}$ ${\displaystyle f\colon A\to B}$ ${\displaystyle g\colon B\to C}$ ${\displaystyle g\circ f\colon A\to C}$ ${\displaystyle \varphi _{g\circ f}=\varphi _{g}\circ \varphi _{f}}$

Coordinate representation

See also : coordinate transformation

This section deals with affine mappings between finite-dimensional affine spaces.

Affine coordinates

If an affine coordinate system has been permanently selected both in the original image space and in the image space , an affine mapping is composed of a linear transformation and a parallel shift with respect to this coordinate system . The linear transformation can then be written as a matrix-vector product and the affine transformation results from the matrix (the mapping matrix ) and the displacement vector : ${\displaystyle {\mathcal {A}}_{1}}$ ${\displaystyle {\mathcal {A}}_{2}}$ ${\displaystyle f}$ ${\displaystyle A}$ ${\displaystyle {\vec {t}}}$

${\displaystyle f({\vec {x}})=A\cdot {\vec {x}}+{\vec {t}}}$

The coordinate vectors and are column vectors in this notation and provide the affine coordinates of the position vectors of a prototype point or an image point. The number of rows of the matrix is equal to the dimension of the space in which is shown (range of values), the number of its columns is equal to the dimension of the depicted space . ${\displaystyle {\vec {x}}}$ ${\displaystyle f({\vec {x}})}$ ${\displaystyle A}$ ${\displaystyle {\mathcal {A}}_{2}}$ ${\displaystyle {\mathcal {A}}_{1}}$

The dimension of the image space of the affine mapping is equal to the rank of the mapping matrix . ${\displaystyle f({\mathcal {A}}_{1})}$ ${\displaystyle A}$

In the case of an affine self-mapping of an affine space, only one affine coordinate system is selected, the coordinate vectors and thus relate to the same coordinate system, the mapping matrix is square, i.e. H. the number of rows and columns is the same. In this context, it is common to identify the affine space with the associated vector space of the displacements. In this sense, the affine self-images include all linear maps (with ), and supplement them with a translation component. ${\displaystyle {\vec {x}}}$ ${\displaystyle f({\vec {x}})}$ ${\displaystyle A}$ ${\displaystyle {\vec {t}}=0}$

An affine self-mapping is an affinity if and only if the determinant of the mapping matrix is not equal to 0. ${\displaystyle A}$

Homogeneous coordinates and extended mapping matrix

→ Main article : Homogeneous coordinates

If homogeneous affine coordinates are selected for the representation both in the original image space and in the image space , then the displacement vector can be combined with the mapping matrix to form an extended mapping matrix : ${\displaystyle {\mathcal {A}}_{1}}$ ${\displaystyle {\mathcal {A}}_{2}}$ ${\displaystyle {\vec {t}}}$ ${\displaystyle A}$ ${\displaystyle A_{\mathrm {erw} }(A,{\vec {t}})}$

${\displaystyle A_{\mathrm {erw} }(A,{\vec {t}})={\begin{pmatrix}A&{\vec {t}}\\{\vec {o}}^{\,T}&1\end{pmatrix}},}$where the transposed zero vector is in the vector space that belongs to. ${\displaystyle {\vec {o}}^{\,T}}$ ${\displaystyle {\mathcal {A}}_{1}}$

The mapping equation then reads for homogeneous coordinate vectors

${\displaystyle f_{h}({\vec {x_{h}}})=A_{\mathrm {erw} }(A,{\vec {t}})\cdot {\vec {x_{h}}}}$.

In this representation of the extended matrix, an additional coordinate is added to the column vector as a homogenizing coordinate : ${\displaystyle x_{n+1}}$ ${\displaystyle {\vec {x}}}$

${\displaystyle {\vec {x}}_{h}={\begin{pmatrix}{\vec {x}}\\1\end{pmatrix}}}$.

This representation through homogeneous coordinates can be interpreted as an embedding of the affine space in a projective space of the same dimension. Then the homogeneous coordinates are to be understood as projective coordinates .

Classification of plane affinities

Affinities are generally differentiated according to how many fixed points they have. This also applies if the affine space has more than two dimensions. A point is a fixed point if it is mapped onto itself through affinity. In the coordinate representation you can determine the coordinate vector of a fixed point by solving the system of equations . Note that fixed points can also exist! ${\displaystyle {\vec {x}}_{p}}$ ${\displaystyle {\vec {x}}_{p}-A\cdot {\vec {x}}_{p}={\vec {t}}}$ ${\displaystyle {\vec {t}}\neq 0}$

1. Axis affinity: A plane affinity in which exactly one straight line remains fixed point by point, it is called the axis of affinity. These include shear , diagonal mirroring (especially vertical axis mirroring) and parallel stretching.
2. Affinity with a center ( central affinity ): an affinity in which exactly one point remains fixed, the center of the affinity. These include the rotation extension (with the special cases centric extension, rotation and point reflection), the shear extension and the Euler affinity. ${\displaystyle Z}$
3. Affinities without a fixed point: These are the pure displacements and successive executions of an axis affinity and a displacement (shear with displacement in a direction different from the axis direction or parallel stretching / oblique reflection with a displacement in the direction of the axis).

The classification is presented in more detail and generalized to higher dimensions in the main article Affinity (Mathematics) .

Normal form of the coordinate representation for plane affinities

A plane affinity is brought to normal form by choosing a suitable affine point base for its coordinate representation. For this purpose, wherever possible, the origin of the coordinate system is placed in a fixed point and the axes of the coordinate system in the direction of fixed lines . The following normal forms hold for affinities in the real affine plane. In the case of a fixed point-free affinity, in addition to the mapping matrix , a displacement vector is required to describe the affinity. ${\displaystyle A}$ ${\displaystyle {\vec {t}}\neq 0}$

1. Axis affinities (the fixed point is the first base point next to the origin ): ${\displaystyle O=E_{0}}$ ${\displaystyle E_{1}}$
   1. Shear ${\displaystyle A={\begin{pmatrix}1&1\\0&1\end{pmatrix}}}$
   2. Oblique mirroring ${\displaystyle A={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}$
   3. Parallel stretching ${\displaystyle A={\begin{pmatrix}1&0\\0&a\end{pmatrix}};\quad a>0}$
2. Central affinities (the fixed point is the origin; the directions of the eigenvectors of the matrix may be chosen as coordinate axes .) ${\displaystyle A}$
   1. Rotary stretching is the stretching factor and the rotation angle, ${\displaystyle A=r\cdot {\begin{pmatrix}\cos(\varphi )&-\sin(\varphi )\\\sin(\varphi )&\cos(\varphi )\end{pmatrix}},\quad }$ ${\displaystyle |r|}$ ${\displaystyle \varphi }$
   2. Shear elongation ${\displaystyle A={\begin{pmatrix}a&1\\0&a\end{pmatrix}};\quad a\in \mathbb {R} \setminus \lbrace 0;1\rbrace }$
   3. Euler affinity ${\displaystyle A={\begin{pmatrix}a&0\\0&b\end{pmatrix}};\quad a\neq b;\quad a,b\in \mathbb {R} \setminus \lbrace 0;1\rbrace .}$

This classification of the affinities also applies more generally to an affine plane to the vector space when is a Euclidean subfield of the real numbers. Here then also applies to the matrix entries: . In the case of rotational stretching, in general - even if the plane is a Euclidean plane with radians - the angle itself is not a body element. ${\displaystyle K^{2}}$ ${\displaystyle K}$ ${\displaystyle a,b,r,\cos(\varphi ),\sin(\varphi )\in K}$ ${\displaystyle \varphi \in \mathbb {R} }$

Special cases

• An affine mapping of a space in itself is called an affine self-mapping . If this self-image is bijective (reversibly unambiguous), it is called affinity .
• An affinity in which every straight line is parallel to its image line is called dilation or homothety . The parallel displacements are special homotheties.
• An affine self-mapping, in which the Euclidean distance of points is retained, is called movement or, especially in the plane case, congruence mapping , such movements are necessarily bijective, i.e. affinities.
• Important affine self-mapping, which are not bijective, are the parallel projections , in which the affine space is mapped onto a real subspace and the restriction is to the identical mapping . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle B}$
• An affine mapping of an affine space in the basic body of this space, which is understood as a one-dimensional affine space above itself, is sometimes referred to as an affine function .

Applications

Graphic applications, computer graphics

Affine images come e.g. B. in cartography and image processing for application.

• Affine images, which are often used in robotics or computer graphics, for example , are rotation (rotation), mirroring , scaling (changing the scale), shear and displacement (translation). All images mentioned are bijective.
• If three-dimensional bodies are to be represented in drawings or graphics - i.e. in two dimensions - non-objective affine images are required.

• This includes the parallel projection with the parallel plans (floor plan, elevation, cross plan; → see normal projection ) as special cases.
• The central projection is generally not an affine mapping. It belongs to the projective mappings , a generalization of the affine mappings.
• In addition, there are other graphic representations that are not based on an affine mapping, for example the Mercator projections for maps .

• Affine images are also used in the standardized description of vector graphics (for example in SVG format ).

Linear transformation in statistics

Affine maps are used as linear transformation in statistical methods , for example .

Distribution parameter of a random variable

A random variable with the expected value and the variance is considered . There is formed a new random variable that a linear transformation is ${\displaystyle X}$ ${\displaystyle \operatorname {E} (X)}$ ${\displaystyle \operatorname {Var} (X)}$ ${\displaystyle X}$

${\displaystyle Y=a+bX,}$

where and are real numbers . ${\displaystyle a}$ ${\displaystyle b}$

The new random variable then has the expected value ${\displaystyle Y}$

${\displaystyle \operatorname {E} (Y)=a+b\operatorname {E} (X),}$

and the variance

${\displaystyle \operatorname {Var} (Y)=b^{2}\operatorname {Var} (X).}$

The following applies in particular: If it is normally distributed , then it is also normally distributed with the above parameters. ${\displaystyle X}$ ${\displaystyle Y}$

example

Let be a random variable with positive variance. A linear transformation is then useful ${\displaystyle X}$

${\displaystyle Y={\frac {X-\operatorname {E} (X)}{\sqrt {\operatorname {Var} (X)}}},}$

because now with and is a so-called standardized random variable. ${\displaystyle Y}$ ${\displaystyle \operatorname {E} (Y)=0}$ ${\displaystyle \operatorname {Var} (Y)=1}$

Distribution parameters of several jointly distributed random variables

Be considered many random variables , . These random variables are summarized in the random vector . The expected values ​​are listed in the expected value vector and the variances and covariances in the covariance matrix . It is a random vector formed of a linear transformation of is ${\displaystyle p}$ ${\displaystyle X_{j}}$ ${\displaystyle j=1,\dots ,p}$ ${\displaystyle {\underline {X}}=(X_{1},\dots ,X_{p})^{T}}$ ${\displaystyle {\underline {\mu }}_{X}}$ ${\displaystyle {\underline {\Sigma }}_{X}}$ ${\displaystyle {\underline {Y}}}$ ${\displaystyle {\underline {X}}}$

${\displaystyle {\underline {Y}}={\underline {a}}+{\underline {B}}\,{\underline {X}},}$

where are a -dimensional column vector and a ( ) - matrix . ${\displaystyle {\underline {a}}}$ ${\displaystyle q}$ ${\displaystyle {\underline {B}}}$ ${\displaystyle q\times p}$

${\displaystyle {\underline {Y}}}$ then has the expectation value vector

${\displaystyle {\underline {\mu }}_{Y}={\underline {a}}+{\underline {B}}\,{\underline {\mu }}_{X}}$

and the covariance matrix

${\displaystyle {\underline {\Sigma }}_{Y}={\underline {B}}\,{\underline {\Sigma }}_{X}\,{\underline {B}}^{T}}$.

The following applies in particular: If -dimensional normally distributed , then -dimensional normally distributed with the above distribution parameters. ${\displaystyle {\underline {X}}}$ ${\displaystyle p}$ ${\displaystyle {\underline {Y}}}$ ${\displaystyle q}$

See also

• Coordinate transformation
• Transformation parameters , 7-parameter transformation

Web links

• "Algebra & Geometry" by Bernhard Kabelka
• Lecture notes "Algebra and Geometry" by Hubert Grassmann (PDF; 891 kB)
• "The affine space" and "affine images" by Joachim Mohr

literature

• Gerd Fischer : Analytical Geometry. 6th edition. Vieweg, Braunschweig / Wiesbaden 1992, ISBN 3-528-57235-3 .
• Hermann Schaal, Ekkehart Glässner: Linear Algebra and Analytical Geometry. Volume 1, Vieweg, Braunschweig / Wiesbaden 1976, ISBN 3-528-03056-9 .
• Uwe Storch, Hartmut Wiebe: Textbook of Mathematics, Volume II: Linear Algebra. BI-Wissenschafts-Verlag, 1990, ISBN 3-411-14101-8 .