Affine space

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Affine_space

In ${\displaystyle {\mathbb{R}}^3,}$ the upper plane (in blue)${\displaystyle P_2}$ is not a vector subspace, since ${\displaystyle \mathbf{0}\notin P_2}$ and ${\displaystyle \mathbf{a}+\mathbf{b}\notin P_2;}$ it is an affine subspace. Its direction (the linear subspace associated with this affine subspace) is the lower (green) plane ${\displaystyle P_1,}$ which is a vector subspace. Although ${\displaystyle \mathbf{a}}$ and ${\displaystyle \mathbf{b}}$ are in ${\displaystyle P_2,}$ their difference is a displacement vector, which does not belong to ${\displaystyle P_2,}$ but belongs to vector space ${\displaystyle P_1.}$

In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments.

In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead displacement vectors, also called translation vectors or simply translations, between two points of the space. Thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point translated from the starting point by that vector.

Any vector space may be viewed as an affine space; this amounts to forgetting the special role played by the zero vector. In this case, elements of the vector space may be viewed either as points of the affine space or as displacement vectors or translations. When considered as a point, the zero vector is called the origin. Adding a fixed vector to the elements of a linear subspace (vector subspace) of a vector space produces an affine subspace. One commonly says that this affine subspace has been obtained by translating (away from the origin) the linear subspace by the translation vector (the vector added to all the elements of the linear space). In finite dimensions, such an affine subspace is the solution set of an inhomogeneous linear system. The displacement vectors for that affine space are the solutions of the corresponding homogeneous linear system, which is a linear subspace. Linear subspaces, in contrast, always contain the origin of the vector space.

The dimension of an affine space is defined as the dimension of the vector space of its translations. An affine space of dimension one is an affine line. An affine space of dimension 2 is an affine plane. An affine subspace of dimension n – 1 in an affine space or a vector space of dimension n is an affine hyperplane.

Informal description

Origins from Alice's and Bob's perspectives. Vector computation from Alice's perspective is in red, whereas that from Bob's is in blue.

The following characterization may be easier to understand than the usual formal definition: an affine space is what is left of a vector space after one has forgotten which point is the origin (or, in the words of the French mathematician Marcel Berger, "An affine space is nothing more than a vector space whose origin we try to forget about, by adding translations to the linear maps"). Imagine that Alice knows that a certain point is the actual origin, but Bob believes that another point—call it p—is the origin. Two vectors, a and b, are to be added. Bob draws an arrow from point p to point a and another arrow from point p to point b, and completes the parallelogram to find what Bob thinks is a + b, but Alice knows that he has actually computed

p + (a − p) + (b − p).

Similarly, Alice and Bob may evaluate any linear combination of a and b, or of any finite set of vectors, and will generally get different answers. However, if the sum of the coefficients in a linear combination is 1, then Alice and Bob will arrive at the same answer.

If Alice travels to

λa + (1 − λ)b

then Bob can similarly travel to

p + λ(a − p) + (1 − λ)(b − p) = λa + (1 − λ)b.

Under this condition, for all coefficients λ + (1 − λ) = 1, Alice and Bob describe the same point with the same linear combination, despite using different origins.

While only Alice knows the "linear structure", both Alice and Bob know the "affine structure"—i.e. the values of affine combinations, defined as linear combinations in which the sum of the coefficients is 1. A set with an affine structure is an affine space.

Definition

An affine space is a set A together with a vector space ${\displaystyle \overrightarrow{A}}$, and a transitive and free action of the additive group of ${\displaystyle \overrightarrow{A}}$ on the set A. The elements of the affine space A are called points. The vector space ${\displaystyle \overrightarrow{A}}$ is said to be associated to the affine space, and its elements are called vectors, translations, or sometimes free vectors.

Explicitly, the definition above means that the action is a mapping, generally denoted as an addition,

${\displaystyle \begin{array}{rl}A\times \overrightarrow{A}& \to A\\ (a,v)& \mapsto a+v,\end{array}}$

that has the following properties.

1. Right identity:

   ${\displaystyle \mathrm{\forall }a\in A,a+0=a}$, where 0 is the zero vector in ${\displaystyle \overrightarrow{A}}$

2. Associativity:

   ${\displaystyle \mathrm{\forall }v,w\in \overrightarrow{A},\mathrm{\forall }a\in A,(a+v)+w=a+(v+w)}$ (here the last + is the addition in ${\displaystyle \overrightarrow{A}}$)

3. Free and transitive action:

   For every ${\displaystyle a\in A}$, the mapping ${\displaystyle \overrightarrow{A}\to A:v\mapsto a+v}$ is a bijection.

The first two properties are simply defining properties of a (right) group action. The third property characterizes free and transitive actions, the onto character coming from transitivity, and then the injective character follows from the action being free. There is a fourth property that follows from 1, 2 above:

4. Existence of one-to-one translations

For all ${\displaystyle v\in \overrightarrow{A}}$, the mapping ${\displaystyle A\to A:a\mapsto a+v}$ is a bijection.

Property 3 is often used in the following equivalent form (the 5th property).

5. Subtraction:

For every a, b in A, there exists a unique ${\displaystyle v\in \overrightarrow{A}}$, denoted b – a, such that ${\displaystyle b=a+v}$.

Another way to express the definition is that an affine space is a principal homogeneous space for the action of the additive group of a vector space. Homogeneous spaces are by definition endowed with a transitive group action, and for a principal homogeneous space such a transitive action is by definition free.

Subtraction and Weyl's axioms

The properties of the group action allows for the definition of subtraction for any given ordered pair (b, a) of points in A, producing a vector of ${\displaystyle \overrightarrow{A}}$. This vector, denoted ${\displaystyle b-a}$ or ${\displaystyle \overrightarrow{ab}}$, is defined to be the unique vector in ${\displaystyle \overrightarrow{A}}$ such that

${\displaystyle a+(b-a)=b.}$

Existence follows from the transitivity of the action, and uniqueness follows because the action is free.

This subtraction has the two following properties, called Weyl's axioms:

1. ${\displaystyle \mathrm{\forall }a\in A,\mathrm{\forall }v\in \overrightarrow{A}}$, there is a unique point ${\displaystyle b\in A}$ such that ${\displaystyle b-a=v.}$
2. ${\displaystyle \mathrm{\forall }a,b,c\in A,(c-b)+(b-a)=c-a.}$

A feature of affine spaces that are Euclidean is the parallelogram property of vectors.

Affine spaces can be equivalently defined as a point set A, together with a vector space ${\displaystyle \overrightarrow{A}}$, and a subtraction satisfying Weyl's axioms. In this case, the addition of a vector to a point is defined from the first of Weyl's axioms.

Affine subspaces and parallelism

An affine subspace (also called, in some contexts, a linear variety, a flat, or, over the real numbers, a linear manifold) B of an affine space A is a subset of A such that, given a point ${\displaystyle a\in B}$, the set of vectors ${\displaystyle \overrightarrow{B}=\{b-a\mid b\in B\}}$ is a linear subspace of ${\displaystyle \overrightarrow{A}}$. This property, which does not depend on the choice of a, implies that B is an affine space, which has ${\displaystyle \overrightarrow{B}}$ as its associated vector space.

The affine subspaces of A are the subsets of A of the form

${\displaystyle a+V=\{a+w:w\in V\},}$

where a is a point of A, and V a linear subspace of ${\displaystyle \overrightarrow{A}}$.

The linear subspace associated with an affine subspace is often called its direction, and two subspaces that share the same direction are said to be parallel.

This implies the following generalization of Playfair's axiom: Given a direction V, for any point a of A there is one and only one affine subspace of direction V, which passes through a, namely the subspace a + V.

Every translation ${\displaystyle A\to A:a\mapsto a+v}$ maps any affine subspace to a parallel subspace.

The term parallel is also used for two affine subspaces such that the direction of one is included in the direction of the other.

Affine map

Given two affine spaces A and B whose associated vector spaces are ${\displaystyle \overrightarrow{A}}$ and ${\displaystyle \overrightarrow{B}}$, an affine map or affine homomorphism from A to B is a map

${\displaystyle f:A\to B}$

such that

${\displaystyle \begin{array}{rl}\overrightarrow{f}:\overrightarrow{A}& \to \overrightarrow{B}\\ b-a& \mapsto f(b)-f(a)\end{array}}$

is a well defined linear map. By ${\displaystyle f}$ being well defined is meant that b – a = d – c implies f(b) – f(a) = f(d) – f(c).

This implies that, for a point ${\displaystyle a\in A}$ and a vector ${\displaystyle v\in \overrightarrow{A}}$, one has

${\displaystyle f(a+v)=f(a)+\overrightarrow{f}(v).}$

Therefore, since for any given b in A, b = a + v for a unique v, f is completely defined by its value on a single point and the associated linear map ${\displaystyle \overrightarrow{f}}$.

Endomorphisms

Main articles: Affine transformation and Affine group

An affine transformation or endomorphism of an affine space ${\displaystyle A}$ is an affine map from that space to itself. One important family of examples is the translations: given a vector ${\displaystyle \overrightarrow{v}}$, the translation map ${\displaystyle T_{\overrightarrow{v}}:A\to A}$ that sends ${\displaystyle a\mapsto a+\overrightarrow{v}}$ for every ${\displaystyle a}$ in ${\displaystyle A}$ is an affine map. Another important family of examples are the linear maps centred at an origin: given a point ${\displaystyle b}$ and a linear map ${\displaystyle M}$, one may define an affine map ${\displaystyle L_{M,b}:A\to A}$ by

$${\displaystyle L_{M,b}(a)=b+M(a-b)}$$

for every ${\displaystyle a}$ in ${\displaystyle A}$.

After making a choice of origin ${\displaystyle b}$, any affine map may be written uniquely as a combination of a translation and a linear map centred at ${\displaystyle b}$.

Vector spaces as affine spaces

Every vector space V may be considered as an affine space over itself. This means that every element of V may be considered either as a point or as a vector. This affine space is sometimes denoted (V, V) for emphasizing the double role of the elements of V. When considered as a point, the zero vector is commonly denoted o (or O, when upper-case letters are used for points) and called the origin.

If A is another affine space over the same vector space (that is ${\displaystyle V=\overrightarrow{A}}$) the choice of any point a in A defines a unique affine isomorphism, which is the identity of V and maps a to o. In other words, the choice of an origin a in A allows us to identify A and (V, V) up to a canonical isomorphism. The counterpart of this property is that the affine space A may be identified with the vector space V in which "the place of the origin has been forgotten".

Relation to Euclidean spaces

Definition of Euclidean spaces

Euclidean spaces (including the one-dimensional line, two-dimensional plane, and three-dimensional space commonly studied in elementary geometry, as well as higher-dimensional analogues) are affine spaces.

Indeed, in most modern definitions, a Euclidean space is defined to be an affine space, such that the associated vector space is a real inner product space of finite dimension, that is a vector space over the reals with a positive-definite quadratic form q(x). The inner product of two vectors x and y is the value of the symmetric bilinear form

${\displaystyle x\cdot y=\frac{1}{2}(q(x+y)-q(x)-q(y)).}$

The usual Euclidean distance between two points A and B is

${\displaystyle d(A,B)=\sqrt{q(B-A)}.}$

In older definition of Euclidean spaces through synthetic geometry, vectors are defined as equivalence classes of ordered pairs of points under equipollence (the pairs (A, B) and (C, D) are equipollent if the points A, B, D, C (in this order) form a parallelogram). It is straightforward to verify that the vectors form a vector space, the square of the Euclidean distance is a quadratic form on the space of vectors, and the two definitions of Euclidean spaces are equivalent.

Affine properties

In Euclidean geometry, the common phrase "affine property" refers to a property that can be proved in affine spaces, that is, it can be proved without using the quadratic form and its associated inner product. In other words, an affine property is a property that does not involve lengths and angles. Typical examples are parallelism, and the definition of a tangent. A non-example is the definition of a normal.

Equivalently, an affine property is a property that is invariant under affine transformations of the Euclidean space.

Affine combinations and barycenter

Let a₁, ..., a_n be a collection of n points in an affine space, and ${\displaystyle {\lambda }_1,\dots ,{\lambda }_n}$ be n elements of the ground field.

Suppose that ${\displaystyle {\lambda }_1+\cdots +{\lambda }_n=0}$. For any two points o and o' one has

${\displaystyle {\lambda }_1\overrightarrow{o{a}_1}+\cdots +{\lambda }_n\overrightarrow{o{a}_n}={\lambda }_1\overrightarrow{o^{\prime }{a}_1}+\cdots +{\lambda }_n\overrightarrow{o^{\prime }{a}_n}.}$

Thus, this sum is independent of the choice of the origin, and the resulting vector may be denoted

${\displaystyle {\lambda }_1{a}_1+\cdots +{\lambda }_n{a}_n.}$

When ${\displaystyle n=2,{\lambda }_1=1,{\lambda }_2=-1}$, one retrieves the definition of the subtraction of points.

Now suppose instead that the field elements satisfy ${\displaystyle {\lambda }_1+\cdots +{\lambda }_n=1}$. For some choice of an origin o, denote by ${\displaystyle g}$ the unique point such that

${\displaystyle {\lambda }_1\overrightarrow{o{a}_1}+\cdots +{\lambda }_n\overrightarrow{o{a}_n}=\overrightarrow{og}.}$

One can show that ${\displaystyle g}$ is independent from the choice of o. Therefore, if

${\displaystyle {\lambda }_1+\cdots +{\lambda }_n=1,}$

one may write

${\displaystyle g={\lambda }_1{a}_1+\cdots +{\lambda }_n{a}_n.}$

The point ${\displaystyle g}$ is called the barycenter of the ${\displaystyle a_i}$ for the weights ${\displaystyle {\lambda }_i}$. One says also that ${\displaystyle g}$ is an affine combination of the ${\displaystyle a_i}$ with coefficients ${\displaystyle {\lambda }_i}$.

Examples

• When children find the answers to sums such as 4 + 3 or 4 − 2 by counting right or left on a number line, they are treating the number line as a one-dimensional affine space.
• The space of energies is an affine space for ${\displaystyle \mathbb{R}}$, since it is often not meaningful to talk about absolute energy, but it is meaningful to talk about energy differences. The vacuum energy when it is defined picks out a canonical origin.
• Physical space is often modelled as an affine space for ${\displaystyle {\mathbb{R}}^3}$ in non-relativistic settings and ${\displaystyle {\mathbb{R}}^{1,3}}$ in the relativistic setting. To distinguish them from the vector space these are sometimes called Euclidean spaces ${\displaystyle \text{E}(3)}$ and ${\displaystyle \text{E}(1,3)}$.
• Any coset of a subspace V of a vector space is an affine space over that subspace.
• If T is a matrix and b lies in its column space, the set of solutions of the equation Tx = b is an affine space over the subspace of solutions of Tx = 0.
• The solutions of an inhomogeneous linear differential equation form an affine space over the solutions of the corresponding homogeneous linear equation.
• Generalizing all of the above, if T : V → W is a linear map and y lies in its image, the set of solutions x ∈ V to the equation Tx = y is a coset of the kernel of T , and is therefore an affine space over Ker T .
• The space of (linear) complementary subspaces of a vector subspace V in a vector space W is an affine space, over Hom(W/V, V). That is, if 0 → V → W → X → 0 is a short exact sequence of vector spaces, then the space of all splittings of the exact sequence naturally carries the structure of an affine space over Hom(X, V).
• The space of connections (viewed from the vector bundle ${\displaystyle E\stackrel{\pi }{\to }M}$, where ${\displaystyle M}$ is a smooth manifold) is an affine space for the vector space of ${\displaystyle \text{End}(E)}$ valued 1-forms. The space of connections (viewed from the principal bundle ${\displaystyle P\stackrel{\pi }{\to }M}$) is an affine space for the vector space of ${\displaystyle \text{ad}(P)}$-valued 1-forms, where ${\displaystyle \text{ad}(P)}$ is the associated adjoint bundle.

Affine span and bases

For any non-empty subset X of an affine space A, there is a smallest affine subspace that contains it, called the affine span of X. It is the intersection of all affine subspaces containing X, and its direction is the intersection of the directions of the affine subspaces that contain X.

The affine span of X is the set of all (finite) affine combinations of points of X, and its direction is the linear span of the x − y for x and y in X. If one chooses a particular point x₀, the direction of the affine span of X is also the linear span of the x – x₀ for x in X.

One says also that the affine span of X is generated by X and that X is a generating set of its affine span.

A set X of points of an affine space is said to be affinely independent or, simply, independent, if the affine span of any strict subset of X is a strict subset of the affine span of X. An affine basis or barycentric frame (see § Barycentric coordinates, below) of an affine space is a generating set that is also independent (that is a minimal generating set).

Recall that the dimension of an affine space is the dimension of its associated vector space. The bases of an affine space of finite dimension n are the independent subsets of n + 1 elements, or, equivalently, the generating subsets of n + 1 elements. Equivalently, {x₀, ..., x_n} is an affine basis of an affine space if and only if {x₁ − x₀, ..., x_n − x₀} is a linear basis of the associated vector space.

Coordinates

There are two strongly related kinds of coordinate systems that may be defined on affine spaces.

Barycentric coordinates

See also: Barycentric coordinate system

Let A be an affine space of dimension n over a field k, and ${\displaystyle \{x_0,\dots ,x_n\}}$ be an affine basis of A. The properties of an affine basis imply that for every x in A there is a unique (n + 1)-tuple ${\displaystyle ({\lambda }_0,\dots ,{\lambda }_n)}$ of elements of k such that

${\displaystyle {\lambda }_0+\cdots +{\lambda }_n=1}$

and

${\displaystyle x={\lambda }_0{x}_0+\cdots +{\lambda }_n{x}_n.}$

The ${\displaystyle {\lambda }_i}$ are called the barycentric coordinates of x over the affine basis ${\displaystyle \{x_0,\dots ,x_n\}}$. If the x_i are viewed as bodies that have weights (or masses) ${\displaystyle {\lambda }_i}$, the point x is thus the barycenter of the x_i, and this explains the origin of the term barycentric coordinates.

The barycentric coordinates define an affine isomorphism between the affine space A and the affine subspace of k^{n + 1} defined by the equation ${\displaystyle {\lambda }_0+\cdots +{\lambda }_n=1}$.

For affine spaces of infinite dimension, the same definition applies, using only finite sums. This means that for each point, only a finite number of coordinates are non-zero.

Affine coordinates

An affine frame of an affine space consists of a point, called the origin, and a linear basis of the associated vector space. More precisely, for an affine space A with associated vector space ${\displaystyle \overrightarrow{A}}$, the origin o belongs to A, and the linear basis is a basis (v₁, ..., v_n) of ${\displaystyle \overrightarrow{A}}$ (for simplicity of the notation, we consider only the case of finite dimension, the general case is similar).

For each point p of A, there is a unique sequence ${\displaystyle {\lambda }_1,\dots ,{\lambda }_n}$ of elements of the ground field such that

${\displaystyle p=o+{\lambda }_1{v}_1+\cdots +{\lambda }_n{v}_n,}$

or equivalently

${\displaystyle \overrightarrow{op}={\lambda }_1{v}_1+\cdots +{\lambda }_n{v}_n.}$

The ${\displaystyle {\lambda }_i}$ are called the affine coordinates of p over the affine frame (o, v₁, ..., v_n).

Example: In Euclidean geometry, Cartesian coordinates are affine coordinates relative to an orthonormal frame, that is an affine frame (o, v₁, ..., v_n) such that (v₁, ..., v_n) is an orthonormal basis.

Relationship between barycentric and affine coordinates

Barycentric coordinates and affine coordinates are strongly related, and may be considered as equivalent.

In fact, given a barycentric frame

${\displaystyle (x_0,\dots ,x_n),}$

one deduces immediately the affine frame

${\displaystyle (x_0,\overrightarrow{x_0{x}_1},\dots ,\overrightarrow{x_0{x}_n})=\left(x_0,x_1-x_0,\dots ,x_n-x_0\right),}$

and, if

${\displaystyle \left({\lambda }_0,{\lambda }_1,\dots ,{\lambda }_n\right)}$

are the barycentric coordinates of a point over the barycentric frame, then the affine coordinates of the same point over the affine frame are

${\displaystyle \left({\lambda }_1,\dots ,{\lambda }_n\right).}$

Conversely, if

${\displaystyle \left(o,v_1,\dots ,v_n\right)}$

is an affine frame, then

${\displaystyle \left(o,o+v_1,\dots ,o+v_n\right)}$

is a barycentric frame. If

${\displaystyle \left({\lambda }_1,\dots ,{\lambda }_n\right)}$

are the affine coordinates of a point over the affine frame, then its barycentric coordinates over the barycentric frame are

${\displaystyle \left(1-{\lambda }_1-\cdots -{\lambda }_n,{\lambda }_1,\dots ,{\lambda }_n\right).}$

Therefore, barycentric and affine coordinates are almost equivalent. In most applications, affine coordinates are preferred, as involving less coordinates that are independent. However, in the situations where the important points of the studied problem are affinely independent, barycentric coordinates may lead to simpler computation, as in the following example.

Example of the triangle

The vertices of a non-flat triangle form an affine basis of the Euclidean plane. The barycentric coordinates allows easy characterization of the elements of the triangle that do not involve angles or distances:

The vertices are the points of barycentric coordinates (1, 0, 0), (0, 1, 0) and (0, 0, 1). The lines supporting the edges are the points that have a zero coordinate. The edges themselves are the points that have one zero coordinate and two nonnegative coordinates. The interior of the triangle are the points whose coordinates are all positive. The medians are the points that have two equal coordinates, and the centroid is the point of coordinates (1/3, 1/3, 1/3).

Change of coordinates

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Case of affine coordinates

Case of barycentric coordinates

Properties of affine homomorphisms

Image and fibers

Let

${\displaystyle f:E\to F}$

be an affine homomorphism, with

${\displaystyle \overrightarrow{f}:\overrightarrow{E}\to \overrightarrow{F}}$

its associated linear map. The image of f is the affine subspace ${\displaystyle f(E)=\{f(a)\mid a\in E\}}$ of F, which has ${\displaystyle \overrightarrow{f}(\overrightarrow{E})}$ as associated vector space. As an affine space does not have a zero element, an affine homomorphism does not have a kernel. However, the linear map ${\displaystyle \overrightarrow{f}}$ does, and if we denote by ${\displaystyle K=\{v\in \overrightarrow{E}\mid \overrightarrow{f}(v)=0\}}$ its kernel, then for any point x of ${\displaystyle f(E)}$, the inverse image ${\displaystyle f^{-1}(x)}$ of x is an affine subspace of E whose direction is ${\displaystyle K}$. This affine subspace is called the fiber of x.

Projection

See also: Projection (mathematics)

An important example is the projection parallel to some direction onto an affine subspace. The importance of this example lies in the fact that Euclidean spaces are affine spaces, and that these kinds of projections are fundamental in Euclidean geometry.

More precisely, given an affine space E with associated vector space ${\displaystyle \overrightarrow{E}}$, let F be an affine subspace of direction ${\displaystyle \overrightarrow{F}}$, and D be a complementary subspace of ${\displaystyle \overrightarrow{F}}$ in ${\displaystyle \overrightarrow{E}}$ (this means that every vector of ${\displaystyle \overrightarrow{E}}$ may be decomposed in a unique way as the sum of an element of ${\displaystyle \overrightarrow{F}}$ and an element of D). For every point x of E, its projection to F parallel to D is the unique point p(x) in F such that

${\displaystyle p(x)-x\in D.}$

This is an affine homomorphism whose associated linear map ${\displaystyle \overrightarrow{p}}$ is defined by

${\displaystyle \overrightarrow{p}(x-y)=p(x)-p(y),}$

for x and y in E.

The image of this projection is F, and its fibers are the subspaces of direction D.

Quotient space

Although kernels are not defined for affine spaces, quotient spaces are defined. This results from the fact that "belonging to the same fiber of an affine homomorphism" is an equivalence relation.

Let E be an affine space, and D be a linear subspace of the associated vector space ${\displaystyle \overrightarrow{E}}$. The quotient E/D of E by D is the quotient of E by the equivalence relation such that x and y are equivalent if

${\displaystyle x-y\in D.}$

This quotient is an affine space, which has ${\displaystyle \overrightarrow{E}/D}$ as associated vector space.

For every affine homomorphism ${\displaystyle E\to F}$, the image is isomorphic to the quotient of E by the kernel of the associated linear map. This is the first isomorphism theorem for affine spaces.

Axioms

Affine spaces are usually studied by analytic geometry using coordinates, or equivalently vector spaces. They can also be studied as synthetic geometry by writing down axioms, though this approach is much less common. There are several different systems of axioms for affine space.

Coxeter (1969, p. 192) axiomatizes the special case of affine geometry over the reals as ordered geometry together with an affine form of Desargues's theorem and an axiom stating that in a plane there is at most one line through a given point not meeting a given line.

Affine planes satisfy the following axioms (Cameron 1991, chapter 2): (in which two lines are called parallel if they are equal or disjoint):

• Any two distinct points lie on a unique line.
• Given a point and line there is a unique line which contains the point and is parallel to the line
• There exist three non-collinear points.

As well as affine planes over fields (or division rings), there are also many non-Desarguesian planes satisfying these axioms. (Cameron 1991, chapter 3) gives axioms for higher-dimensional affine spaces.

Purely axiomatic affine geometry is more general than affine spaces and is treated in a separate article.

Relation to projective spaces

An affine space is a subspace of a projective space, which is in turn the quotient of a vector space by an equivalence relation (not by a linear subspace)

Affine spaces are contained in projective spaces. For example, an affine plane can be obtained from any projective plane by removing one line and all the points on it, and conversely any affine plane can be used to construct a projective plane as a closure by adding a line at infinity whose points correspond to equivalence classes of parallel lines. Similar constructions hold in higher dimensions.

Further, transformations of projective space that preserve affine space (equivalently, that leave the hyperplane at infinity invariant as a set) yield transformations of affine space. Conversely, any affine linear transformation extends uniquely to a projective linear transformation, so the affine group is a subgroup of the projective group. For instance, Möbius transformations (transformations of the complex projective line, or Riemann sphere) are affine (transformations of the complex plane) if and only if they fix the point at infinity.

Affine algebraic geometry

In algebraic geometry, an affine variety (or, more generally, an affine algebraic set) is defined as the subset of an affine space that is the set of the common zeros of a set of so-called polynomial functions over the affine space. For defining a polynomial function over the affine space, one has to choose an affine frame. Then, a polynomial function is a function such that the image of any point is the value of some multivariate polynomial function of the coordinates of the point. As a change of affine coordinates may be expressed by linear functions (more precisely affine functions) of the coordinates, this definition is independent of a particular choice of coordinates.

The choice of a system of affine coordinates for an affine space ${\displaystyle {\mathbb{A}}_k^n}$ of dimension n over a field k induces an affine isomorphism between ${\displaystyle {\mathbb{A}}_k^n}$ and the affine coordinate space kⁿ. This explains why, for simplification, many textbooks write ${\displaystyle {\mathbb{A}}_k^n=k^n}$, and introduce affine algebraic varieties as the common zeros of polynomial functions over kⁿ.

As the whole affine space is the set of the common zeros of the zero polynomial, affine spaces are affine algebraic varieties.

Ring of polynomial functions

By the definition above, the choice of an affine frame of an affine space ${\displaystyle {\mathbb{A}}_k^n}$ allows one to identify the polynomial functions on ${\displaystyle {\mathbb{A}}_k^n}$ with polynomials in n variables, the ith variable representing the function that maps a point to its ith coordinate. It follows that the set of polynomial functions over ${\displaystyle {\mathbb{A}}_k^n}$ is a k-algebra, denoted ${\displaystyle k\left[{\mathbb{A}}_k^n\right]}$, which is isomorphic to the polynomial ring ${\displaystyle k\left[X_1,\dots ,X_n\right]}$.

When one changes coordinates, the isomorphism between ${\displaystyle k\left[{\mathbb{A}}_k^n\right]}$ and ${\displaystyle k[X_1,\dots ,X_n]}$ changes accordingly, and this induces an automorphism of ${\displaystyle k\left[X_1,\dots ,X_n\right]}$, which maps each indeterminate to a polynomial of degree one. It follows that the total degree defines a filtration of ${\displaystyle k\left[{\mathbb{A}}_k^n\right]}$, which is independent from the choice of coordinates. The total degree defines also a graduation, but it depends on the choice of coordinates, as a change of affine coordinates may map indeterminates on non-homogeneous polynomials.

Zariski topology

See also: Zariski topology

Affine spaces over topological fields, such as the real or the complex numbers, have a natural topology. The Zariski topology, which is defined for affine spaces over any field, allows use of topological methods in any case. Zariski topology is the unique topology on an affine space whose closed sets are affine algebraic sets (that is sets of the common zeros of polynomial functions over the affine set). As, over a topological field, polynomial functions are continuous, every Zariski closed set is closed for the usual topology, if any. In other words, over a topological field, Zariski topology is coarser than the natural topology.

There is a natural injective function from an affine space into the set of prime ideals (that is the spectrum) of its ring of polynomial functions. When affine coordinates have been chosen, this function maps the point of coordinates ${\displaystyle \left(a_1,\dots ,a_n\right)}$ to the maximal ideal ${\displaystyle ⟨X_1-a_1,\dots ,X_n-a_n⟩}$. This function is a homeomorphism (for the Zariski topology of the affine space and of the spectrum of the ring of polynomial functions) of the affine space onto the image of the function.

The case of an algebraically closed ground field is especially important in algebraic geometry, because, in this case, the homeomorphism above is a map between the affine space and the set of all maximal ideals of the ring of functions (this is Hilbert's Nullstellensatz).

This is the starting idea of scheme theory of Grothendieck, which consists, for studying algebraic varieties, of considering as "points", not only the points of the affine space, but also all the prime ideals of the spectrum. This allows gluing together algebraic varieties in a similar way as, for manifolds, charts are glued together for building a manifold.

Cohomology

Like all affine varieties, local data on an affine space can always be patched together globally: the cohomology of affine space is trivial. More precisely, ${\displaystyle H^i\left({\mathbb{A}}_k^n,\mathbf{F}\right)=0}$ for all coherent sheaves F, and integers ${\displaystyle i>0}$. This property is also enjoyed by all other affine varieties. But also all of the étale cohomology groups on affine space are trivial. In particular, every line bundle is trivial. More generally, the Quillen–Suslin theorem implies that every algebraic vector bundle over an affine space is trivial.

Smallholding

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Smallholding

Woman smallholder farmers in Kenya. In many parts of Africa and other parts of the world, women are the primary smallholders. In many contexts, women face unequal access to land, markets, knowledge, and other assets needed to maintain their farms.

Small vegetable farm in Hainan, China

A smallholding or smallholder is a small farm operating under a small-scale agriculture model. Definitions vary widely for what constitutes a smallholder or small-scale farm, including factors such as size, food production technique or technology, involvement of family in labor and economic impact. Smallholdings are usually farms supporting a single family with a mixture of cash crops and subsistence farming. As a country becomes more affluent, smallholdings may not be self-sufficient, but may be valued for the rural lifestyle. As the sustainable food and local food movements grow in affluent countries, some of these smallholdings are gaining increased economic viability. There are an estimated 500 million smallholder farms in developing countries of the world alone, supporting almost two billion people.

Small-scale agriculture is often in tension with industrial agriculture, which finds efficiencies by increasing outputs, monoculture, consolidating land under big agricultural operations, and economies of scale. Certain labor-intensive cash-crops, such as cocoa production in Ghana or Côte d'Ivoire, rely heavily on small holders; globally, as of 2008 90% of cocoa is grown by smallholders. These farmers rely on cocoa for up to 60 to 90 per cent of their income. Similar trends in supply chains exist in other crops like coffee, palm oil, and bananas. In other markets, small scale agriculture can increase food system investment in small holders improving food security. Today some companies try to include smallholdings into their value chain, providing seed, feed or fertilizer to improve production.

Because smallholding farms frequently require less industrial inputs and can be an important way to improve food security and sustainable food systems in less-developed contexts, addressing the productivity and financial sustainability of small holders is an international development priority and measured by indicator 2.3 of Sustainable Development Goal 2. Additionally, since agriculture has such large impacts on climate change, Project Drawdown described "Sustainable Intensification for Smallholders" an important method for climate change mitigation.

Issues

Productivity

Agriculture

According to conventional theory, economies of scale allow agricultural productivity, in terms of inputs versus outputs, to rise as the size of the farm rises. Specialization has also been a major factor in increasing agricultural productivity, for example as commodity processing began to move off the farm in the 19th century, farmers could spend more effort on primary food production.

Although numerous studies show that larger farms are more productive than smaller ones, some writers state that whilst conventional farming creates a high output per worker, some small-scale, sustainable, polyculture farmers can produce more food per acre of land.

A smallholder coffee farmer in Colombia contributing her coffee to an agricultural cooperative. Cooperatives give small farmers an opportunity to be more competitive in markets, especially markets like coffee and cocoa where many of the purchasers are large businesses with high market power.

Small farms have some economic advantages. Farmers support the local economy of their communities. An American study showed that small farms with incomes of $100,000 or less spend almost 95 percent of their farm-related expenses within their local communities. The same study took into comparison the fact that farms with incomes greater than $900,000 spend less than 20 percent of their farm-related expenses in the local economy.

Small-scale agriculture often sells products directly to consumers. Disintermediation gives the farmer the profit that would otherwise go to the wholesaler, the distributor, and the supermarket. About two-thirds of the revenue is expended on product marketing. If farmers sell their products directly to consumers, they receive a higher percentage of the retail price, although they will spend more time selling the same amounts of product, which is an opportunity cost.

Food security

Because smallholding farms frequently require less industrial inputs and can be an important way to improve food security in less-developed contexts, addressing the productivity and financial sustainability of small holders is an international development priority and measured by indicator 2.3 of Sustainable Development Goal 2. The International Fund for Agricultural Development has an ongoing program for Adaptation for Smallholder Agriculture.

During the global COVID-19 pandemic, and the attendant disruptions of food systems, their role has become more important.

Environmental and climate adaptation

While the historical focus on smallholders has been increasing global food supply under climate change and the role played by smallholder communities, climate adaptation efforts are still hindered by lack of information on how smallholder farmers are experiencing and responding to climate change. There is lack of detailed, context-specific information on what climate change portends to smallholder farmers in different and widely varying agroecological environments and socio-economic realities, and what management strategies they are employing to deal with these impacts.

Especially for smallholders working in commodity crops, climate change introduces an increasing amount of variability to farmer economic viability; for example, coffee production globally is under increased threat, and smallholders in East Africa, such as in the Ugandan, Tanzanian or Kenyan industries, are rapidly losing both viable coffee land and productivity of plants.

In some cases, smallholders are an important source of deforestation. For example, smallholders are an important component of the oil palm industry of Southeast Asia, contributing 40% of the production. Because such farmers are less able to access financing than larger businesses, they are unable to fund methods to increase the productivity of their farms when yields decline, increasing their need to clear more land. Increasing productivity, especially amongst smallholder farms, is an important way to decrease the amount of land needed for farming and slow environmental degradation through processes like deforestation.

Formats

The definition of a small farm has varied over time and by country. Agricultural economists have analyzed the distinctions among farm sizes since the field's inception. Traditional agricultural economic theory considered small farms inefficient, a stance that began to be challenged in the 1950s. An overview of research published by the World Bank in 1998 indicated that the productivity of small farms often exceeded that of larger ones.

Hobby farms

A hobby farm (also called a lifestyle block in New Zealand, or acreage living or rural residential in Australia) is a smallholding or small farm that is maintained without expectation of being a primary source of income. Some are held merely to provide recreational land for horses or other use. Others are managed as working farms for secondary income, or are even run at an ongoing loss as a lifestyle choice by people with the means to do so, functioning more like a country home than a business.

Nucleus estate and smallholder

Nucleus estate and smallholder (NES) is a farming system for commodity crops, often oil palm, practised in different world regions. It is most famous today for its application in the palm oil sector in Indonesia. The nucleus is the part of such a plantation that is under concession and management of the company, while another part of the plantation is operated by smallholders typically on their own land but planted by the company. NES farming is a particular form of contract farming.

Croft

The Shetland Crofthouse Museum at Dunrossness, Shetland, with peat stacked outside

A croft is a traditional Scottish term for a fenced or enclosed area of land, usually small and arable, and usually, but not always, with a crofter's dwelling thereon. A crofter is one who has tenure and use of the land, typically as a tenant farmer, especially in rural areas.

Developing countries

In many developing countries, smallholding is a small plot of land with low rental value, used to grow crops. By some estimates, there are 525 million smallholder farmers in the world. These farms vary in land sizes, production and labor intensities. The distribution of farm sizes depends on a number of agroecological and demographic conditions, as well as on economic and technological factors. Smallholders are critical to local and regional food systems, as well as livelihoods, and especially so during periods of food supply chain disruptions. Smallholders dominate production in certain key sectors such as coffee and cocoa. Various types of agribusinesses enterprises work with smallholding farmers in a range of roles including buying crops, providing seed, and acting as financial institutions.

In low-income countries, women make up 43 percent of smallholding agricultural labor but produce 60–80 percent of food crops.

India

In India, there is five sizes classification for smallholders. These are 'marginal' less than 1 hectare (2+1⁄2 acres), 'small' between 1 and 2 hectares (2+1⁄2 and 5 acres), 'semi medium' between 2 and 4 hectares (5 and 10 acres), 'medium' between 4 and 10 hectares (9.9 and 24.7 acres), 'large' above 10 hectares (25 acres). If we use 4 hectares (10 acres) (marginal + small + medium) as a threshold, 94.3% of holdings are small and these constitute 65.2% of all farmland. The bulk of India's hungry and poor people are constituted of smallholder farmers and landless people. 78% country's farmers own less than 2 hectares (5 acres), which constitutes 33% of total farmland but at the same time, they produce 41% of the country's food grains. 20% of the world's poor live in India, although the country was self-sufficient in food production in 2002 due to the first Green Revolution started in the latter half of the twentieth century, numerous households lacked resources to purchase food. Holdings less than 2 ha contributed 41% of total food grain production in 1991 compared to 28% in 1971, which means a substantial increase, whereas medium holdings registered a mere 3% increase in the same period and large holdings registered a decline from 51 to 35%. This signifies the importance of smallholders in the Green Revolution and the attainment of national food security. Smallholder families are becoming more vulnerable and more disadvantaged due to the expansion of international trade liberalisation. The needs and aspirations of small farmers must feature prominently in policies of market reform that seek to improve food and nutritional security. India's total increase rate of productivity across the farming sector was far less in 1990's when compared to previous decades.

Kenya

Kenyan smallholder dairy farmer

Kenya's smallholder means someone who owns, possess or produces agricultural products in small-scale . smallholder production accounts for 78 percent of total agricultural production and 70 percent of commercial production. Majority of the smallholder population work in farm sizes averaging 0.47 hectares (11⁄4 acres). This represents the vast majority of Kenya's rural poor population who depend on agriculture for their livelihood. Adverse risk events during the period 1980–2012 led to production losses in smallholder farms resulting in a drop in agricultural gross domestic product (GDP) of 2 percent or more. Increasing the productivity of smallholder farmers is encouraged due to its potential of improving food availability, increasing rural incomes, lowering poverty rates, and growing the economy. Diversification of crops in smallholder farms is one of the potential strategies in sustaining agricultural productivity, and copping with marketing risks. It is also a transitional step from subsistence to commercial agriculture. Age, education of household head, type of crops, cropping system, amount of credit, and irrigation facilities are some of the factors influencing diversification in smallholder farms.

Tanzania

Along the upper and middle reaches of the Nduruma River in the Pangani River Basin, Tanzania, there is not enough water to go around. Smallholder farmers address inequities in land and water distribution by enforcing existing traditional local rules. Whilst larger estate farms may have governmental licences guaranteeing rights to the water, a study found that those large-scale farms which adhere to the traditional water rights structures fare better in terms of social reputation, which better ensures their access to water. Adhering to the water law in order to enforce their permits is less effective, as regional Tanzanian local governments generally attempt to avoid conflict with their populace. On a larger scale, however, existing traditional rules are ineffective in maintaining cooperation among users along the Nduruma River.

Thailand

Further information: Agriculture in Thailand

In 1975, there were 4.2 million smallholder farming households in Thailand. In 2013, Thailand had 5.9 million smallholder farming households. The average area of these smallholdings had shrunk from 3.7 to 3.2 hectares (9+1⁄4 to 8 acres) over that period. Instead of farms getting larger and less numerous, as has been the case in the Global North, the reverse happened: they got smaller and more numerous.

United States

Several definitions of small farm have been formulated in legislation. In 1977 the US Congress, via the Food and Agriculture Act of 1977, defined a small farm as one with sales under $20,000. At the time these comprised 70% of farms in the US. The Act sponsored additional research on small farming operations by US land grant universities and their extension services and mandated that an annual report on these activities be issued by the US Secretary of Agriculture. A 1997 study by the United States Small Farms Commission defined small farms as those with less than $250,000 in gross receipts annually on which day-to-day labor and management are provided by the farmer and/or the farm family that owns the production, or owns or leases the productive assets. In 2000, such farms accounted for about 90% of the more than 2.1 million U.S. farms, but only about 40% of U.S. farm production.

The concentration of production on fewer and larger operations is a longstanding concern among some segments of the agricultural community. Others view these changes as inevitable, and even necessary to maintain the efficiency and competitiveness of the sector.

Farm typology analysis by the USDA Economic Research Service divides the small family farm category into five groups:

1. limited-resource farms;
2. retirement farms;
3. residential/lifestyle farms;
4. farming occupation/lower-sales,
5. farming occupation/high-sales.

Technology for small farmers

Main article: Right to repair

Many farmers are upset by their inability to fix the new types of high-tech farm equipment. This is due mostly to companies using intellectual property law to prevent farmers from having the legal right to fix their equipment (or gain access to the information to allow them to do it). This has encouraged groups such as Open Source Ecology and Farm Hack to begin to make open-source agricultural machinery.

European Union

The debate concerning the role of small farms within the European Union is ongoing. The European Commission states that more than three quarters of farm holdings in the European Union are less than 10 hectares, with a large number less than five hectares, although as of 2009 it had not established a formal definition of the term that could be used in its Common Agricultural Policy. The public perception of the possible benefits of small-scale farming has led to requests for further studies from the European Commission.