Affine Spaces and Translations

Introduction

A vector space has a distinguished point, its origin, and every linear statement is made with respect to it. An affine space is what remains when the origin is forgotten: a set of points on which the additive group of a vector space acts freely and transitively, so that any two points determine a difference vector but no point is singled out. This is a torsor structure, and it is the correct language for geometry, since points, lines, planes and barycentres are all affine notions and none of them survives the loss of translational symmetry that a fixed origin would impose.

Throughout, $F$ is a field, $V$ is an $F$-vector space of dimension $n$, and an affine space with direction $V$ is a set $\mathbb{A}$ equipped with a free and transitive action of $(V,+)$. Translating between the affine and linear pictures is done by choosing an origin, and every construction in the article is checked to be independent of that choice. The affine group $\operatorname{Aff}(V)=V \rtimes \operatorname{GL}(V)$ is the symmetry group of the affine structure, exactly as $\operatorname{GL}(V)$ is the symmetry group of the linear structure. The Euclidean case, where the group is $V \rtimes O(V,Q)$, is treated in Part IV.

Affine Spaces

Definition as a Torsor

Definition. Let $V$ be a vector space. An affine space with direction $V$ is a set $\mathbb{A}$ together with a map

$$ \mathbb{A} \times V \longrightarrow \mathbb{A}, \qquad (a,v) \longmapsto a+v, $$

such that (i) $a+0=a$ and $(a+v)+w=a+(v+w)$ for all $a \in \mathbb{A}$, $v,w \in V$; (ii) for every $a,b \in \mathbb{A}$ there is exactly one $v \in V$ with $a+v=b$. The unique such $v$ is written $b-a$, and $\mathbb{A}$ is called a torsor under $V$. Its elements are points, and $V$ is the direction space.

Condition (i) says that $V$ acts on $\mathbb{A}$; condition (ii) says the action is free (if $a+v=a$ then $v=0$) and transitive. Free and transitive actions are called simply transitive.

Proposition. With the notation above, for all points $a,b,c$:

(i) $a-a=0$ and $b-a=-(a-b)$;

(ii) $(c-b)+(b-a)=c-a$;

(iii) $a+(b-a)=b$.

Proof. All three are immediate from the uniqueness in the definition: the element $c-a$ is the unique $v$ with $a+v=c$, and evaluating the left-hand sides at $a$ gives $c$.

Choosing an Origin

Definition. An origin in $\mathbb{A}$ is a point $o \in \mathbb{A}$. Given $o$, the position map is

$$ \varphi_o:\mathbb{A} \longrightarrow V, \qquad a \longmapsto a-o . $$

Proposition. For every origin $o$, the position map $\varphi_o$ is a bijection. For two origins $o,o'$,

$$ \varphi_{o'}(a)=\varphi_o(a)-(o'-o), $$

so the two identifications of $\mathbb{A}$ with $V$ differ by the translation $v \mapsto v-(o'-o)$. Consequently any statement about $\mathbb{A}$ that is to be intrinsic must be invariant under the translations of $V$.

Proof. The map is a bijection because for each $v$ there is a unique $a=o+v$ with $a-o=v$. The relation $\varphi_{o'}(a)=a-o'=(a-o)-(o'-o)$ gives the second statement.

The content of the definition is thus that an affine space is a vector space together with a forgotten origin, and the choice of an origin converts it into a vector space in a way that is canonical only up to translation. This is why statements in affine geometry come in two forms: an intrinsic form using differences of points, and a coordinate form after an origin is chosen.

Example. The set of solutions of a consistent system $Ax=b$ is an affine space with direction $\ker A$, by the rank criterion of the linear-maps article: if $x_0$ is one solution, every solution is $x_0+u$ with $Au=0$.

Translations

The Translation Group

Definition. For $v \in V$, the translation by $v$ is the map

$$ t_v:\mathbb{A} \longrightarrow \mathbb{A}, \qquad t_v(a)=a+v . $$

Proposition. (i) $t_0=\operatorname{id}_{\mathbb{A}}$ and $t_v \circ t_w=t_{v+w}$, so $v \mapsto t_v$ is an injective homomorphism $(V,+) \to \operatorname{Sym}(\mathbb{A})$; the image $T(\mathbb{A})=\{t_v\}$ is the translation group, isomorphic to $(V,+)$ and normal in the group of all affine automorphisms.

(ii) For $v \neq 0$, the translation $t_v$ has no fixed point.

(iii) No translation $t_v$ with $v \neq 0$ is linear, for any choice of origin making $\mathbb{A}=V$: in that identification $t_v(x)=x+v$ and $t_v(0)=v \neq 0$.

Proof. (i) is the action axiom. (ii) $t_v(a)=a$ means $a+v=a$, hence $v=0$ by freeness. (iii) A linear map sends $0$ to $0$, and $t_v(0)=v \neq 0$.

Part (iii) is the precise sense in which translations are not linear maps but affine maps: they preserve the affine structure and not the linear structure. The failure is exactly the displacement of the origin, and it is not repaired by a change of origin, since changing the origin leaves a translation unchanged; conjugation by a general affine map $(T,b)$ carries $t_v$ to $t_{Tv}$, so the translations form a normal subgroup isomorphic to $V$.

Simply Transitive Action

Proposition. The translation group acts simply transitively on $\mathbb{A}$. Consequently, after the choice of any point $o$, the map $t_v \mapsto o+v$ is a bijection $T(\mathbb{A}) \to \mathbb{A}$.

Proof. Transitivity is axiom (ii): for any $a,b$ there is $v$ with $b=a+v=t_v(a)$. Freeness is (ii) as well: if $t_v(a)=a$ then $v=0$.

The Affine Group

Definition and Composition

Definition. An affine automorphism of $\mathbb{A}$ is a bijection $f:\mathbb{A} \to \mathbb{A}$ such that $f(a+v)=f(a)+L(v)$ for a linear map $L \in \operatorname{GL}(V)$ and all $a,v$. The group of affine automorphisms is the affine group $\operatorname{Aff}(\mathbb{A})=\operatorname{Aff}(V)$.

Proposition. Affine automorphisms are exactly the maps of the form

$$ f(a)=o'+T(a-o) $$

for some origin $o \in \mathbb{A}$, some $T \in \operatorname{GL}(V)$ and some $o' \in \mathbb{A}$. The linear part $T$ is independent of $o$, and $\operatorname{Aff}(V) \cong V \rtimes \operatorname{GL}(V)$, where the semidirect product is taken with respect to the natural action of $\operatorname{GL}(V)$ on $V$.

Proof. Choose an origin $o$ and identify $\mathbb{A}$ with $V$ by $a \mapsto a-o$. Then an affine automorphism becomes a map $x \mapsto Tx+b$ with $T \in \operatorname{GL}(V)$ and $b \in V$, and every such map is an affine automorphism. Composition is

$$ (x \mapsto T_1x+b_1)\circ(x \mapsto T_2x+b_2)=(x \mapsto T_1T_2x+T_1b_2+b_1), $$

which is the multiplication law $(T_1,b_1)(T_2,b_2)=(T_1T_2,\,T_1b_2+b_1)$ of the semidirect product $V \rtimes \operatorname{GL}(V)$ with $\operatorname{GL}(V)$ acting on $V$ in the standard way. Changing the origin conjugates the pair by a translation from the translation subgroup $V$, so the decomposition is intrinsic.

In the notation $(T,b)$ for the map $x \mapsto Tx+b$, the exact sequence

$$ 1 \longrightarrow V \longrightarrow \operatorname{Aff}(V) \longrightarrow \operatorname{GL}(V) \longrightarrow 1 $$

is split by $T \mapsto (T,0)$, and the action of $\operatorname{GL}(V)$ on the normal translation subgroup $V$ is the defining representation. For $n \ge 1$ the affine group is generated by the translations and the linear maps: every element is the product $(T,b)=(I,b)(T,0)$ of a translation and a linear map, while the opposite product is $(T,0)(I,b)=(T,Tb)$, so the order of the factors is not interchangeable.

Invariants

Proposition. $\operatorname{Aff}(V)$ acts transitively on $\mathbb{A}$, and the stabiliser of a point $o$ is the subgroup $\{(T,0)\} \cong \operatorname{GL}(V)$ of linear maps fixing $o$. The action is $2$-transitive for $n \ge 1$: it is transitive on ordered pairs of distinct points.

Proof. Transitivity: $(0,b)$ sends $o$ to $o+b$, so the translations already act transitively. The stabiliser of $o$ consists of the maps $x \mapsto Tx$ with $b=0$, a copy of $\operatorname{GL}(V)$. For $2$-transitivity, given distinct $a,b$ and distinct $a',b'$, choose $T \in \operatorname{GL}(V)$ with $T(b-a)=b'-a'$, and then a translation correcting the images of $a$.

Affine Combinations and Independence

Barycentres

Definition. Let $a_1,\dots,a_k \in \mathbb{A}$ and $\lambda_1,\dots,\lambda_k \in F$ with $\sum_i\lambda_i=1$. The affine combination $\sum_i\lambda_ia_i$ is the point

$$ o+\sum_{i=1}^{k}\lambda_i(a_i-o), $$

which is independent of the origin $o$. When all $\lambda_i=1/k$ and $k$ is invertible in $F$, this point is the barycentre of $a_1,\dots,a_k$.

Proposition. The affine combination is well defined, and it is the unique point $b$ such that $\sum_i\lambda_i(b-a_i)=0$ in the sense of the difference map.

Proof. Independence of the origin: for $o'$ another origin,

$$ o'+\sum_i\lambda_i(a_i-o')=o+\sum_i\lambda_i(a_i-o)+(o'-o)\Bigl(1-\sum_i\lambda_i\Bigr)=o+\sum_i\lambda_i(a_i-o), $$

using $\sum_i\lambda_i=1$. The characterisation is immediate from the definition.

The requirement $\sum\lambda_i=1$ is what replaces the condition for a linear combination to be well defined without a chosen zero; a linear combination of points has no intrinsic meaning. Over a field of characteristic $p$ and a set of $p$ points, the barycentre with weights $1/p$ does not exist, and one uses instead any weights summing to $1$ that avoid the characteristic.

Affine Independence and Bases

Definition. Points $a_0,\dots,a_k \in \mathbb{A}$ are affinely independent if the vectors $a_1-a_0,\dots,a_k-a_0$ are linearly independent in $V$. An affine basis of $\mathbb{A}$ is a set of $n+1$ affinely independent points.

Proposition. $a_0,\dots,a_k$ are affinely independent if and only if the only relation $\sum_i\lambda_ia_i=0$ in the sense of affine combinations with $\sum_i\lambda_i=0$ is the trivial one $\lambda_0=\cdots=\lambda_k=0$. Every affine basis $a_0,\dots,a_n$ gives every point $b \in \mathbb{A}$ a unique expression

$$ b=\sum_{i=0}^{n}\alpha_ia_i, \qquad \sum_{i=0}^{n}\alpha_i=1, $$

the barycentric coordinates $(\alpha_0,\dots,\alpha_n)$ of $b$ with respect to the basis.

Proof. Translating by $a_0$ turns the vectors $a_i-a_0$ into a basis of $V$, and $b-a_0$ has a unique expression in that basis; the coefficients together with $\alpha_0=1-\sum_{i\ge1}\alpha_i$ give the statement.

Affine Subspaces

Definition and Incidence

Definition. An affine subspace of $\mathbb{A}$ is a subset of the form

$$ B=a+W=\{a+w : w \in W\} $$

for a point $a$ and a linear subspace $W \subseteq V$, called the direction of $B$. The dimension of $B$ is $\dim_F W$.

Proposition. (i) The direction of a nonempty affine subspace is determined by the set: $W=\{b-c : b,c \in B\}$. (ii) If $B=a+W$ and $C=b+U$ then $B \subseteq C$ if and only if $W \subseteq U$ and $a-b \in U$. (iii) Two affine subspaces with the same direction are either equal or disjoint. (iv) The intersection of two affine subspaces is either empty or an affine subspace with direction $W \cap U$. Two affine subspaces are parallel when $W \subseteq U$ or $U \subseteq W$; disjointness alone does not force parallelism, since the skew lines of a three-dimensional space are disjoint and neither direction contains the other.

Proof. (i) The differences of elements of $a+W$ are exactly the elements of $W$. (ii) If $B\subseteq C$ then $a \in C$, so $a-b \in U$, and for $w \in W$ one has $a+w \in C$, whence $w=(a+w)-a \in U$ and $W \subseteq U$; conversely $W \subseteq U$ and $a-b \in U$ give $a+W \subseteq b+U$. (iii) If $a+W$ and $b+W$ meet, then $a-b \in W$ by (ii) applied both ways, and the two are equal. (iv) If $c \in B \cap C$ then $B=c+W$, $C=c+U$, and $B \cap C=c+(W \cap U)$.

For $n=1$ an affine subspace of dimension $1$ is the whole of $\mathbb{A}$, and of dimension $0$ a single point: over a one-dimensional direction space there is nothing else. For $n=2$ the subspaces of dimension $1$ are the lines, for $n=3$ the subspaces of dimension $1$ and $2$ are the lines and planes, and in general a hyperplane is an affine subspace of dimension $n-1$, of the form $\{a: \varphi(a-o)=\lambda\}$ for a nonzero linear functional $\varphi$. Two distinct lines in a plane meet in a point or are disjoint and parallel, and this is the intersection statement (iv) with $\dim W=\dim U=1$: either $W=U$ (parallel or equal) or $W \cap U=0$ and the intersection is a single point.

Affine Maps

Definition. An affine map $f:\mathbb{A} \to \mathbb{A}'$ between affine spaces with direction spaces $V,V'$ is a map such that $f(a+v)=f(a)+L(v)$ for a linear map $L:V \to V'$ and all $a,v$. The linear map $L$ is the linear part of $f$.

Proposition. Affine maps are exactly the maps $f(a)=o'+L(a-o)$ for some origins $o,o'$; they form a set $\operatorname{Aff}(\mathbb{A},\mathbb{A}')=\operatorname{Hom}_F(V,V') \times V'$ under the correspondence $(L,b) \leftrightarrow (x \mapsto Lx+b)$ after origins are chosen, and composition is composition of maps. Affine automorphisms are the case $L$ invertible.

Proof. The computation is the one already done in coordinates for $\operatorname{Aff}(V)$: composing $x \mapsto L_1x+b_1$ with $x \mapsto L_2x+b_2$ gives $x \mapsto L_1L_2x+L_1b_2+b_1$.

An affine map with linear part zero is constant; an affine map with $L=\operatorname{id}$ is a translation; an affine map fixing a point $o$ and with $L$ invertible is linear in the coordinates centred at $o$.

Summary

An affine space with direction a vector space $V$ is a set $\mathbb{A}$ on which $V$ acts freely and transitively; equivalently, every ordered pair of points has a unique difference in $V$, and the difference map satisfies $(c-b)+(b-a)=c-a$. Choosing an origin identifies $\mathbb{A}$ with $V$, and two origins differ by a translation, so an affine space is a vector space with its origin forgotten.

The translations $t_v(a)=a+v$ form a normal subgroup isomorphic to $(V,+)$ and act simply transitively; a nonzero translation has no fixed point and is not linear. The affine group $\operatorname{Aff}(\mathbb{A})=\operatorname{Aff}(V)\cong V \rtimes \operatorname{GL}(V)$ consists of the maps $x \mapsto Tx+b$ with $T$ invertible, and sits in the split exact sequence $1 \to V \to \operatorname{Aff}(V) \to \operatorname{GL}(V) \to 1$; it acts transitively, and the stabiliser of a point is a copy of $\operatorname{GL}(V)$.

Affine combinations $\sum\lambda_ia_i$ with $\sum\lambda_i=1$ are well defined independently of the origin and give barycentres when the weights are equal and the number of points is invertible; affinely independent points give barycentric coordinates, and an affine basis of $n+1$ points gives every point a unique coordinate vector summing to $1$. Affine subspaces $a+W$ have a well-defined direction, two subspaces with the same direction are equal or disjoint, and a nonempty intersection has direction $W \cap U$; in dimensions two and three this is the incidence theory of points, lines and planes. Affine maps are the maps with a linear part, and affine automorphisms form the affine group.

The Euclidean group $V \rtimes O(V,Q)$ and the classification of its isometries are treated in Part IV.

Summary of Notation

Symbol	Meaning
$F$	a field
$V$	direction vector space, $\dim_F V=n$
$\mathbb{A}$	affine space with direction $V$
$a+v$, $b-a$	action of $V$ on $\mathbb{A}$ and difference of points
$o$	an origin; $\varphi_o(a)=a-o$
$t_v$	translation by $v$, $t_v(a)=a+v$
$T(\mathbb{A}) \cong (V,+)$	translation group
$\operatorname{Aff}(\mathbb{A})=\operatorname{Aff}(V)$	affine group
$V \rtimes \operatorname{GL}(V)$	semidirect product structure, $(T,b)(T',b')=(TT',Tb'+b)$
$(T,b)$	affine map $x \mapsto Tx+b$
$\sum_i\lambda_ia_i$, $\sum_i\lambda_i=1$	affine combination
$(\alpha_0,\dots,\alpha_n)$	barycentric coordinates
$B=a+W$	affine subspace with direction $W$
$Q$, $g$	quadratic form and its polar form