Affine transformation

In geometry and linear algebra, areas of mathematics, an affine transformation is (also affine transformation above, especially at a bijective mapping ) a mapping between two affine spaces are preserved in the collinearity, parallelism and tissue conditions or become irrelevant. More precisely:

A bijective affine transformation of an affine space onto itself is called affinity.

In school mathematics and some applications (for example, in statistics, see below) special affine mappings are also called linear map or linear function. Generally, however, mathematical parlance, a linear mapping is a homomorphism of vector spaces.

• 3.1 Normal form of the coordinate representation of plane affinities

• 5.1 Graphical applications, computer graphics
• 5.2 Linear transformation in statistics 5.2.1 distribution parameters of a random variable X
• 5.2.2 distribution parameters from multiple jointly distributed random variables

Definition

A mapping between affine spaces and is called affine if there is a linear map between the associated vector spaces, so that

Applies to all points. This call and the connection vectors of the pre-image or the image points.

Coordinate representation

This section deals with affine mappings between finite dimensional affine spaces. If an affine coordinate system has been chosen both in the prototype space and in image space, then sits down with respect to this coordinate system and an affine transformation of a linear transformation and a parallel shift together. The linear transformation can then be written as a matrix-vector product and the affine transformation arises from the matrix ( the imaging matrix ) and the displacement vector:

The coordinate vectors and are in this notation column vectors and represent the affine coordinates of the position vectors of a prototype point or a pixel represents the number of rows of the matrix is equal to the dimension of space is mapped into the ( range of values ​​), the number of its columns is equal to the dimension of the space depicted.

The dimension of the image space of the affine transformation is equal to the rank of the projection matrix.

In an affine self-map of an affine space only an affine coordinate system is selected, the coordinate vectors and thus refer to the same coordinate system, the projection matrix is square, ie its row and column number are equal. In this connection, it is common to identify the affine space with the corresponding displacements of the vector space. In this sense, the affine self-maps include all linear maps ( with ), and complement, a translation component.

An affine self-map is accurate then an affinity, if the determinant of the transformation matrix is not 0.

Homogeneous coordinates and Advanced imaging matrix

If you choose to display both the original image space and in image space homogeneous affine coordinates, you can be the displacement vector with the projection matrix to summarize an extended projection matrix:

The mapping equation then for homogeneous coordinate vectors

In this representation of the augmented matrix, an additional coordinate of the column vector is added as homogenizing coordinate:

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

Classification of plane affinities

Affinities are generally distinguished first by how many fixed points they have. This is true even if the affine space has more than two dimensions. A point is fixed point, if it is mapped by the affinity to itself. In the coordinate display, one can determine the coordinates of a fixed point vector by solving the system of equations. Note that can exist for fixed points!

Detail and generalized to higher dimensions, the classification in the main article affinity ( mathematics) is presented.

Normal form of the coordinate representation of plane affinities

A flat affinity is brought to normal form by selecting an appropriate base to their point affine coordinate representation. To this end, wherever that is possible to set the origin of the coordinate system to a fixed point and the axes of the coordinate system in the direction of Fixgeraden. The following normal forms apply to affinities in the real affine plane. In the case of fixed-point- free affinity is out of the imaging matrix nor a displacement vector to describe the affinity needed.

This classification also applies to the affinities at a general affine plane to the vector space if a Euclidean part of the real number is. Here then is in addition to the matrix entries. For three dilations in general - even if the plane is a Euclidean plane with radian - is the angle itself is not a body element.

Special cases

• An affine transformation of a space into itself is called a self- affine mapping. Is this self-mapping is bijective ( bijective ), it is called affinity.
• An affinity in which each parallel straight lines to their image, is called dilation or homothety. The parallel shifts are special homotheties.
• A self- affine mapping, wherein the Euclidean distance between points is maintained, ie, movement, or, particularly in the case of a plane, congruence, such movements are necessary bijective so affinities.
• Important affinity self-images that are not bijective, the parallel projections in which the affine space is mapped to a real sub-space and is the limitation on the identity mapping.
• An affine transformation of an affine space in the main body of this space, which is regarded in this context as one-dimensional affine space over itself, it sometimes referred to as affine function.

Applications

Graphical applications, computer graphics

Affine transformations are, for example, in cartography and image editing application.

• Affine transformations, which are often, for example in robotics and computer graphics are used, are rotation (rotation), mirroring, scaling ( change of scale ), and shear displacement (translation). All these pictures are bijective.
• If three-dimensional body drawing or graphically - so in two dimensions - should be displayed, nichtbijektive affine transformations are needed.

• This includes the parallel projection with the parallel cracks (ground plan, elevation, cross plan; → see normal projection ) as special cases.
• The central projection is not affine in general. It belongs to the projective pictures, a generalization of affine mappings.
• There are other graphical representations, which no affine transformation is based on, for example for maps Mercator projections.

• In the standardized description of vector graphics also affine transformations are used (for example, in SVG format ).

Linear transformation in statistics

As a linear transformation, affine transformations are used for example in the statistical methods.

Distribution parameters of a random variable X

Consider a random variable with the expected value and the variance. There is formed a new random variable, which is a linear transformation of,

Where and are real numbers.

The new random variable is then the expected value

And variance

Especially applies: If normally distributed, so is normally distributed with the above parameters.

Let X be a random variable with positive variance. Useful is then a linear transformation

For now is, and a so-called standardized random variable.

Distribution parameters from multiple jointly distributed random variables

Considered many random variables. It summarizes these random variables together in the random vector. The expectation values ​​are in the expected value vector and lists the variances and covariances in the covariance matrix. There is formed a random vector that is a linear transformation of,

With a - dimensional column vector and a ( ) matrix () are.

Then has the expectation value vector

And the covariance matrix

Applies in particular: Is - dimensional normally distributed, so -dimensionally is normally distributed with the above distribution parameters.