Affine coordinate system

Affine coordinates are coordinates that are associated with a point one -dimensional affine space with respect to a so-called affine base point in the mathematical subfield of linear algebra, which is an ordered set of points in space with certain properties (see later in this article).

One then distinguishes inhomogeneous affine coordinates, the most common form in which the coordinates of a point is an ordered set ( tuples) of numbers, and homogeneous forms in which these coordinates constitute a tuple.

Using the affine coordinate system described herein, an affine matrix can be represented by a figure, will be explained in the article affine.

Affine coordinates are closely related to part ratios: affine coordinates can be converted in part ratios and vice versa. This relationship is described in the article " split ratio ", notably in the section "Sub- ratio and affine coordinates".

In synthetic affine geometry coordinates for affinity levels are introduced by geometric construction that coordinate structure. Here are points of a fixed chosen precisely the level as affine coordinates. Affine levels over a body of this concept leads to the same geometric ( inhomogeneous ) affine coordinates, such as the approach described in this article from the analytic geometry. → Refer to the affine coordinates in synthetic geometry of the main article " Ternärkörper ".

• 2.1 Numerical Example
• 2.2 linear equation
• 2.3 Systems of Equations

Definitions

Affine coordinate system in the standard model

An affine subspace of a - dimensional vector space over the field - the standard model of the -dimensional affine space - has the shape with a vector and a subspace is. The subspace is uniquely determined by ( and is called the subspace corresponding to ) not the vector, however, it can be selected from any. The dimension is defined as the dimension of.

Is one - dimensional affine space, so called an affine basis points, if the vectors form a basis of the vector space.

In this case, for every uniquely determined by and.

Inhomogeneous and homogeneous barycentric coordinate affine

In an affine subspace, there is no excellent zero point. An affine basis takes this into account. If one chooses a basis vector of any such, so is a basis of the corresponding subspace. Is of no importance, as is, that is there with. It follows

Substituting, so true and. In this representation, the basis points are equal again, none of the points is kind of excellent.

The coordinates are called inhomogeneous affine coordinates are called barycentric coordinates of affinity relative to the base. The barycentric coordinates provide in contrast to the non-homogeneous coordinates and then formally the same representation of the point when the vector is not the zero vector of the vector space.

As a homogeneous affine coordinates are meaning the tuple. ( In the literature it is also often used.) Here, the -dimensional affine point space with the hyperplane with the equation is identified in the vector space. It is this homogeneous coordinate vectors but also be regarded as points of the projective space. Then describe the coordinates for the same affine point describe distant points ( directions ) of the affine space. The presentation by homogeneous coordinates can be used, among other things, to describe arbitrary affine transformations with a (extended) mapping matrix without translation vector (→ to this coordinate representation see main article Homogeneous coordinates to advanced imaging matrix see Affine Figure: Advanced imaging matrix ).

Affine coordinate system in affine space point

An affine point space is a set together with a vector space of so-called translations and a figure that a displacement vector assigns to each pair of points in such a way that certain properties are met ( see Article affine space ). This approach makes it particularly clear that no point is awarded.

An affine base ( also affine point basis) of is then a set of points, so that the vectors form a basis of. The point is the origin of the affine coordinate system, which is determined by this base point. The dimension is called the dimension of. The dimension of is if and only if there is an affine base points.

The affine space with and is an example, and it can be shown that this dimension is the most common affinity to affine space, wherein an affinity between two areas and an image is affine to the linear transformation is a bijective with all there.

If we choose an affine basis, so there is exactly one affinity, with the canonical basis of was. If now, the affine coordinates of with respect to the affine basis in the affine space can be calculated as above. These numbers are called the affine coordinates of with respect to the affinity by certain coordinate system. The affinity is also referred to as affine coordinate system; which is the underlying concept that the coordinates carries from to. In this conception is the origin and the coordinate representation of the position vector of a point.

Examples

Numerical example

Be the real three-dimensional coordinate space. Then form the three points ( 1,0,0 ), ( 0,1,0 ) and ( 0,0,1 ) together with the origin (0,0,0) is an affine basis. For a point, the numbers are the affine coordinates with respect to this basis.

One chooses the affinity of the base origin and the points, and so are the affine coordinates of a point, where, as it is

Linear equation

Lines are one-dimensional affine subspaces, and two distinct points form an affine basis. The representation of the points in affine coordinates leads to the linear equation in the form of so-called parameter, because it is

Systems of equations

The amount of solution of an inhomogeneous linear system of equations forms an affine space. Is a special solution to the inhomogeneous equation system, and a base of the solution space of the associated homogeneous system, thus forming a basis of the affinity -affinity solution space of the non-homogeneous equation system. To each solution there is therefore uniquely determined by and. This view shows the known fact that there are no excellent special solution for an inhomogeneous system of linear equations.

Convex combinations

A convex combination of points is a special representation in barycentric affine coordinates, in which not only but also applies to all.