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affine geometry (Definition)

Affine Subspaces

Let $ V$ be a vector space over a division ring $ D$. An affine subspace $ A$ of $ V$ is defined to be a coset of a (linear) vector subspace of $ V$. In other words, $ A=S+v$, where $ S$ is a subspace of $ V$ and $ v\in V$ is a vector.

It is evident that $ A$ is uniquely determined by $ S$, and $ v$ up to translation by a vector in $ S$. Suppose $ A=S_1+v_1=S_2+v_2$. Since $ 0\in S_1\cap S_2$, we see that $ v_1-v_2\in S_1\cap S_2$. Thus we may assume that $ S_1+v=S_2+v$ for any vector $ v\equiv v_1\equiv v_2\pmod{S}$, where $ S$ is the subspace $ S_1\cap S_2$. If $ w\in S_1$, then $ w+v\in S_2+v$, and hence $ w\in S_2$. So $ S_1\subseteq S_2$. By symmetry, this shows $ S_1=S_2=S$. So we may speak of $ S$ as the associated linear space of $ A$.

The dimension of an affine subspace $ A=S+v$ is defined to be the dimension of $ S$. If $ A$ has dimension 0, it is called a point. $ A$ is a line or a plane if it has dimension 1 or 2, respectively. General affine subspaces are also called flats, or subspaces when there is no confusion with the associated (linear) subspaces. An affine hyperplane is an affine subspace whose associated linear space is a hyperplane in $ V$.

Affine Geometry

Affine geometry is, generally speaking, the study the geometric properties of affine subspaces. In particular, it is the study of the incidence structure on affine subspaces. Operationally, we may define an affine geometry $ \mathcal{A}(V)$ of a vector space $ V$ to be the poset of all affine subspaces of $ V$, orderd by set theoretic inclusion. Points in $ \mathcal{A}(V)$ are commonly written without the set theoretic brackets, so that $ v\in\mathcal{A}(V)$ means $ \lbrace v\rbrace \in\mathcal{A}(V)$.

Next, we can define an incidence relation $ I$ on $ \mathcal{A}(V)$ so that $ (A,B)\in I$ iff $ A\subseteq B$ or $ B\subseteq A$. Together with $ I$, $ \mathcal{A}(V)$ becomes an incidence geometry. Two flats $ A$ and $ B$ are said to be parallel if they have the same associated subspace. As a result, two parallel flats are never incident unless they are equal. Also, given a point $ v\in \mathcal{A}(V)$ not incident with $ A$, we can always find a flat $ B$ incident with $ v$ and parallel to $ A$. If $ A=S+w$ with $ w\neq v$, simply take $ B=S+v$. This makes $ \mathcal{A}(V)$ an affine incidence geometry.

In addition, we define $ A\vee B$ to be the smallest flat in $ \mathcal{A}(V)$ that contains both $ A$ and $ B$. By Zorn's lemma, $ A\vee B$ exists. Since $ A\vee B$ is also unique, $ \vee$ is well-defined. This turns $ \mathcal{A}(V)$ into an upper semilattice. If $ S_1$ is the associated subspace of $ A$ and $ S_2$ is the associated subspace of $ B$, then $ \operatorname{span}(S_1\cup S_2)$ is the associated subspace of $ A\vee B$. The definition of $ \vee$ can be extended to an arbitrary set of flats, so that $ \bigvee \mathcal{S}$ is the smallest flat that contains all flats in $ \mathcal{S}\subseteq\mathcal{A}$. In fact, it is not hard to see that $ \mathcal{A}(V)$ is complete semilattice.

However, since $ A\cap B$ may be empty, $ \mathcal{A}(V)$ is not a lattice in general via the “meet” ($ =\cap$) operation. But when $ A\cap B\neq \varnothing$, $ A\cap B\in \mathcal{A}(V)$. So $ \cap$ is a partially defined operator on $ \mathcal{A}(V)$. If one adjoins the empty set $ \varnothing$ to $ \mathcal{A}(V)$, then $ \mathcal{A}(V)$ becomes a lattice. $ \varnothing$ is called the null subspace and its dimension is defined to be $ -1$. One can show that $ \mathcal{A}(V)$ is a geometric lattice.

Although $ 0\in \mathcal{A}(V)$, it is not special, since all points are treated equally; there is no notion of an origin in $ \mathcal{A}(V)$. The notion of a metric is also absent, since the underlying vector space is not assumed to have an inner product. In fact, perpendicularity is not defined in $ \mathcal{A}(V)$. In contrast, both notions are important in Euclidean geometry, where an inner product has been defined, so that 0 is the unique vector with 0 length.

Affine versus Projective

Affine geometry and projective geometry are intimately related. Given an affine geometry $ \mathcal{A}(V)$ one can construct projective geometries. One easy way is to identify flats that are parallel to each other. Because the parallel relation $ \parallel$ is an equivalence relation, we can partition $ \mathcal{A}(V)$ into equivalence classes. Since each equivalence class is represented by exactly one subspace $ S$ of $ V$, so $ \mathcal{A}(V)/\parallel$ can be identified with $ PG(V)$. Of course, $ PG(V)$ can also be viewed as a sub-poset of $ \mathcal{A}(V)$ (simply by taking all the subspaces of $ V$ in $ \mathcal{A}(V)$). More generally, if we fix any point $ v\in \mathcal{A}(V)$, and take all flats that are incident with $ v$, the resulting subset $ PG_v(V)$ forms a modular complemented geometric lattice that is isomorphic to $ PG(V)$. In fact, $ PG_v(V)$ has the structure of a projective geometry.

Another way to construct a projective geometry from an affine one is to adjoin extra elements to $ \mathcal{A}(V)$. Remember that $ \mathcal{A}(V)$ itself is not a lattice, but simply adjoining $ \varnothing$ to $ \mathcal{A}(V)$ won't give us a projective geometry either, because the resulting lattice is not modular (take two parallel lines $ \ell_1,\ell_2$ and a point $ P$ lying on $ \ell_1$; then $ P\vee (\ell_2\wedge \ell_1)=P$, while $ (P\vee\ell_2)\wedge \ell_1=\ell_1$). We start by taking a vector space $ U$ such that $ V$ is a subspace of $ U$ of codimension 1 (This can be done by linear algebra). Our objective is to show that $ \mathcal{A}(V)$ is embeddable in $ PG(U)$.

Let $ u\in U$ be a non-zero vector and look at the affine hyperplane $ V+u$. Each affine subspace of the form $ S+v$ in $ V$ has the form $ S+v+u$ in $ V+u$, where $ S$ is a subspace of $ V$ and $ v\in V$. Let $ \mathcal{A}_u(V)$ the collection of all affine subspaces $ S+v+u$ (affine subspaces of $ U$ that are incident with $ V+u$). There is an obvious one-to-one order preserving correspondence between $ \mathcal{A}(V)$ and $ \mathcal{A}_u(V)$.

Next, every affine subspace $ S+v+u$ is the intersection of $ V+u$ and a subspace $ W$ of $ U$ such that $ S+v+u\subseteq W$ and $ \operatorname{dim}(W)= \operatorname{S}+1$. Just take $ W=\operatorname{span}(S,v+u)$. Clearly $ S+v+u\subseteq W\cap V+u$. In addition, $ v+u\notin S\subseteq V$, or else $ u\in V$ gives us a contradiction. So $ \operatorname{dim}(W)= \operatorname{S}+1$. Finally, if $ x\in V+u\cap W$, then $ x=y+u=s+k(v+u)$, where $ y\in V$, $ s\in S$, and $ k\in D$. So $ x\equiv u\equiv ku \pmod V$. This implies $ (1-k)u\in V$. But $ u\notin V$, $ k=1$. Therefore $ x=s+u+v\in S+v+u$. This means that $ W\cap V+u=S+v+u$.

The above paragraph shows there is a one-to-one order preserving map from $ \mathcal{A}(V)$ to $ PG(U)$. If we delete all subspaces of $ V$ from $ PG(U)$, and call

$\displaystyle PG(U)/PG(V)=\lbrace W\in PG(U)\mid W$ is not a subspace of $\displaystyle V\rbrace,$
then we actually get an order-preserving bijection between $ \mathcal{A}(V)$ and $ PG(U)/PG(V)$.



"affine geometry" is owned by CWoo. [ full author list (2) ]
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See Also: affine combination, affine transformation

Other names:  associated subspace
Also defines:  flat, associated linear subspace, affine subspace
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Cross-references: bijection, map, implies, contradiction, subspace, intersection, order, one-to-one, obvious, collection, non-zero vector, linear algebra, codimension, parallel lines, structure, isomorphic, complemented, modular, subset, fix, equivalence classes, partition, equivalence relation, relation, projective geometry, length, Euclidean geometry, perpendicularity, inner product, metric, origin, geometric lattice, null, empty set, operator, operation, lattice, complete semilattice, upper semilattice, well-defined, Zorn's lemma, contains, addition, affine incidence geometry, incident, parallel, incidence geometry, iff, incidence relation, inclusion, poset, incidence structure, properties, hyperplane, plane, line, point, dimension, symmetry, translation, vector, vector subspace, coset, division ring, vector space
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This is version 12 of affine geometry, born on 2006-06-09, modified 2006-11-03.
Object id is 7988, canonical name is AffineGeometry.
Accessed 4972 times total.

Classification:
AMS MSC: 51A15 (Geometry :: Linear incidence geometry :: Structures with parallelism)
 51A45 (Geometry :: Linear incidence geometry :: Incidence structures imbeddable into projective geometries)
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