# Affine coordinates

Affine coordinates are coordinates that are assigned in the mathematical subfield of linear algebra to a point of a -dimensional affine space with respect to a so-called affine point base, that is an ordered set of points of the space with certain properties (see further below in this article). ${\ displaystyle n}$${\ displaystyle n + 1}$

A distinction is then made between inhomogeneous affine coordinates , the most common form in which the coordinates of a point is an ordered set ( tuple ) of numbers, and homogeneous forms in which these coordinates form a tuple. ${\ displaystyle n}$${\ displaystyle n + 1}$

With the aid of the affine coordinate systems described here, an affine mapping can be represented by a mapping matrix.

Affine coordinates are closely related to partial ratios : Affine coordinates can be converted into partial ratios and vice versa.

In synthetic geometry , affine coordinates for affine planes are introduced through a geometric construction, the coordinate construction . Points of a fixed straight line of the plane serve as affine coordinates. For affine planes over a body , this geometric concept leads to the same (inhomogeneous) affine coordinates as the analytical geometry procedure described in this article . → See the main article " Ternary bodies " for the affine coordinates in synthetic geometry .

## Definitions

### Affine coordinate system in the standard model

Let be an affine space with an associated - vector space . Be the dimension of . ${\ displaystyle A}$${\ displaystyle K}$ ${\ displaystyle V}$${\ displaystyle n}$${\ displaystyle V}$

Then points are called an affine basis if the vectors form a basis of . ${\ displaystyle n + 1}$${\ displaystyle p_ {0}, \ dotsc, p_ {n}}$${\ displaystyle p_ {1} -p_ {0}, \ dotsc, p_ {n} -p_ {0}}$${\ displaystyle V}$

In this case there are clearly certain with and for each . ${\ displaystyle p \ in A}$${\ displaystyle \ lambda _ {0}, \ dotsc, \ lambda _ {n} \ in K}$${\ displaystyle p = \ lambda _ {0} p_ {0} + \ dotsb + \ lambda _ {n} p_ {n}}$${\ displaystyle \ lambda _ {0} + \ dotsb + \ lambda _ {n} = 1}$

The notation means that the equation in applies to one (and therefore every) point . ${\ displaystyle p = \ lambda _ {0} p_ {0} + \ dotsb + \ lambda _ {n} p_ {n}}$${\ displaystyle o \ in A}$${\ displaystyle po = \ lambda _ {0} (p_ {0} -o) + \ dotsb + \ lambda _ {n} (p_ {n} -o)}$${\ displaystyle V}$

### Inhomogeneous, barycentric and homogeneous affine coordinates

There is no marked zero point in affine space . An affine base takes this into account. If one chooses a basis vector arbitrarily, for example , then is a basis of the associated vector space. For each one has so unique with . It follows ${\ displaystyle A}$${\ displaystyle p_ {0}, \ dotsc, p_ {n}}$${\ displaystyle p_ {0}}$${\ displaystyle p_ {1} -p_ {0}, \ dotsc, p_ {n} -p_ {0}}$${\ displaystyle p \ in A}$${\ displaystyle \ mu _ {1}, \ dotsc, \ mu _ {n} \ in K}$${\ displaystyle p-p_ {0} = \ mu _ {1} (p_ {1} -p_ {0}) + \ dotsb + \ mu _ {n} (p_ {n} -p_ {0})}$

${\ displaystyle p = p_ {0} + \ mu _ {1} (p_ {1} -p_ {0}) + \ dotsb + \ mu _ {n} (p_ {n} -p_ {0}) = \ left (1- \ sum _ {i = 1} ^ {n} \ mu _ {i} \ right) p_ {0} + \ mu _ {1} p_ {1} + \ dotsb + \ mu _ {n} p_ {n}}$

If you set

${\ displaystyle \ lambda _ {0} = 1- \ sum _ {i = 1} ^ {n} \ mu _ {i}}$, ,${\ displaystyle \ lambda _ {1} = \ mu _ {1}, \ dotsc, \ lambda _ {n} = \ mu _ {n}}$

so applies and . In this representation, the base points are again equal; none of the points is distinguished in any way. ${\ displaystyle p = \ lambda _ {0} p_ {0} + \ dotsb + \ lambda _ {n} p_ {n}}$${\ displaystyle \ lambda _ {0} + \ dotsb + \ lambda _ {n} = 1}$${\ displaystyle p_ {0}, \ dotsc, p_ {n}}$

The coordinates are called inhomogeneous affine coordinates, called barycentric affine coordinates of with respect to the base . In contrast to the inhomogeneous coordinates, the barycentric coordinates also formally provide the same representation of the point if the vector is not the zero vector of the vector space. ${\ displaystyle (\ mu _ {1}; \ dotsc; \ mu _ {n}) \ in K ^ {n}}$${\ displaystyle (\ lambda _ {0}; \ dotsc; \ lambda _ {n}) \ in K ^ {n + 1}}$${\ displaystyle p}$${\ displaystyle p_ {0}, \ dotsc, p_ {n}}$${\ displaystyle p}$${\ displaystyle p_ {0}}$

As homogeneous affine coordinate is defined as the tuple . (Also used frequently in the literature ). This notation is motivated by the interpretation of the -dimensional affine point space as the given subset of the projective space . In projective space has the "homogeneous" induced coordinates, all with how the same point describe it for so may set. The representation using homogeneous coordinates can be used, among other things, to describe any affine mappings with an (extended) mapping matrix without a translation vector (→ for this coordinate representation see main article Homogeneous coordinates , for the extended mapping matrix see Affine mapping: Extended mapping matrix ). ${\ displaystyle n + 1}$${\ displaystyle (\ mu _ {1}; \ dotsc; \ mu _ {n}; 1) \ in K ^ {n + 1}}$${\ displaystyle (1; \ mu _ {1}; \ dotsc; \ mu _ {n}) \ in K ^ {n + 1}}$${\ displaystyle n}$${\ displaystyle x_ {n + 1} \ not = 0}$ ${\ displaystyle KP ^ {n}}$${\ displaystyle K ^ {n + 1}}$${\ displaystyle (r \ cdot \ mu _ {1}; \ dotsc; r \ cdot \ mu _ {n}; r \ cdot \ mu _ {n + 1}) \ in K ^ {n + 1}}$${\ displaystyle r \ in K \ setminus \ lbrace 0 \ rbrace}$${\ displaystyle (\ mu _ {1}; \ dotsc; \ mu _ {n}; \ mu _ {n + 1}) \ in K ^ {n + 1} \ setminus \ lbrace 0 \ rbrace}$${\ displaystyle \ mu _ {n + 1} \ not = 0}$${\ displaystyle \ mu _ {n + 1} = 1}$

For an affine basis there is exactly one affinity with , where the canonical basis of is. If now , the affine coordinates of with respect to the affine base in the affine space can be calculated as above. The affinity is also called the affine coordinate system ; this is based on the idea that the coordinates carry from to . In this view, the origin and the coordinate representation is the position vector of a point . ${\ displaystyle p_ {0}, \ dotsc, p_ {n} \ in A}$${\ displaystyle f \ colon K ^ {n} \ rightarrow A}$${\ displaystyle f (0) = p_ {0}, f (e_ {1}) = p_ {1}, \ dotsc, f (e_ {n}) = p_ {n}}$${\ displaystyle e_ {1}, \ dotsc, e_ {n}}$${\ displaystyle K ^ {n}}$${\ displaystyle p \ in A}$${\ displaystyle f ^ {- 1} (P) \ in K ^ {n}}$${\ displaystyle 0, e_ {1}, \ dotsc, e_ {n}}$${\ displaystyle K ^ {n}}$${\ displaystyle f \ colon K ^ {n} \ rightarrow A}$${\ displaystyle f}$${\ displaystyle K ^ {n}}$${\ displaystyle A}$${\ displaystyle f (0)}$${\ displaystyle f ^ {- 1} (P)}$${\ displaystyle P}$

## Examples

### Numerical example

Let be the three-dimensional real coordinate space . Then the three points and together with the origin form an affine basis. For a point , the numbers are the affine coordinates with respect to this base. ${\ displaystyle A = \ mathbb {R} ^ {3}}$${\ displaystyle (1,0,0), (0,1,0)}$${\ displaystyle (0,0,1)}$${\ displaystyle (0,0,0)}$${\ displaystyle (x, y, z) \ in \ mathbb {R} ^ {3}}$${\ displaystyle x, y, z}$

If one chooses the affine basis from the origin and the points , and , then the affine coordinates for a point are given by, because it holds ${\ displaystyle (1,0,0)}$${\ displaystyle (0,1,0)}$${\ displaystyle (-1,1,1)}$${\ displaystyle \ lambda, \ mu, \ nu}$${\ displaystyle (x, y, z) \ in \ mathbb {R} ^ {3}}$${\ displaystyle \ lambda = x + z, \ \ mu = yz, \ \ nu = z}$

${\ displaystyle (x + z) {\ begin {pmatrix} 1 \\ 0 \\ 0 \ end {pmatrix}} + (yz) {\ begin {pmatrix} 0 \\ 1 \\ 0 \ end {pmatrix}} + z {\ begin {pmatrix} -1 \\ 1 \\ 1 \ end {pmatrix}} = {\ begin {pmatrix} x \\ y \\ z \ end {pmatrix}}.}$

### Straight line equation

Straight lines are one-dimensional affine subspaces and two different points each form an affine basis. The representation of the points of in affine coordinates leads to the straight line equation in the so-called parametric form , because it is ${\ displaystyle g}$${\ displaystyle p_ {0}, p_ {1} \ in g}$${\ displaystyle g}$

${\ displaystyle g = \ {\ lambda p_ {0} + \ mu p_ {1} | \, \ lambda, \ mu \ in \ mathbb {R}, \ lambda + \ mu = 1 \} = \ {(1 - \ mu) p_ {0} + \ mu p_ {1} | \, \ mu \ in \ mathbb {R} \} = \ {p_ {0} + \ mu (p_ {1} -p_ {0}) | \, \ mu \ in \ mathbb {R} \}}$.

### Systems of equations

The set of solutions of an inhomogeneous system of linear equations forms an affine space. If there is a special solution of the inhomogeneous system of equations and a basis of the solution space of the associated homogeneous system, then form an affine basis of the affine solution space of the inhomogeneous system of equations. For each solution there are clearly certain ones with and . This observation shows the well-known fact that there is no excellent special solution for an inhomogeneous system of linear equations. ${\ displaystyle p_ {0}}$${\ displaystyle u_ {1}, \ dotsc, u_ {n}}$${\ displaystyle p_ {0}, p_ {1} = p_ {0} + u_ {1}, \ dotsc, p_ {n} = p_ {0} + u_ {n}}$${\ displaystyle p}$${\ displaystyle \ lambda _ {0}, \ dotsc, \ lambda _ {n} \ in K}$${\ displaystyle p = \ lambda _ {0} p_ {0} + \ dotsb + \ lambda _ {n} p_ {n}}$${\ displaystyle \ lambda _ {0} + \ dotsb + \ lambda _ {n} = 1}$

## Convex combinations

A convex combination of points is a special representation in barycentric affine coordinates that applies not only to but also to all . ${\ displaystyle n + 1}$${\ displaystyle p_ {0}, \ dotsc, p_ {n}}$${\ displaystyle \ lambda _ {0}, \ dotsc, \ lambda _ {n} \ in \ mathbb {R}}$${\ displaystyle \ lambda _ {0} + \ dotsb + \ lambda _ {n} = 1}$${\ displaystyle \ lambda _ {i} \ geq 0}$${\ displaystyle i = 1, \ dotsc, n}$

## literature

• Gerd Fischer : Analytical Geometry (= Rororo-Vieweg 35). Rowohlt, Reinbek near Hamburg 1978, ISBN 3-499-27035-8 .
• Hermann Schaal, Ekkehart Glässner: Linear Algebra and Analytical Geometry. Volume 1. Vieweg, Braunschweig 1976, ISBN 3-528-03056-9 .
• Uwe Storch , Hartmut Wiebe: Textbook of Mathematics. For mathematicians, computer scientists and physicists. Volume 2: Linear Algebra. BI-Wissenschafts-Verlag, Mannheim 1990, ISBN 3-411-14101-8 .