Affine space · aiwiki.page

An affine space is a structure in geometry that retains the displacement operations of a vector space without specifying an origin. Its elements are points: subtracting two points produces a vector, and adding a vector to a point produces another point. Addition of arbitrary points and scalar multiplication of individual points are not intrinsically defined. This distinction allows geometric constructions to be expressed independently of the choice of coordinate origin. (people.math.harvard.edu)

Definition and displacement

Let $V$ be a vector space over a field $K$ . An affine space modeled on $V$ is a nonempty set $A$ equipped with an operation

A\times V\longrightarrow A,\qquad (p,v)\longmapsto p+v,

satisfying

p+0=p,\qquad (p+v)+w=p+(v+w),

and the requirement that, for every $p,q\in A$ , there is exactly one $v\in V$ with $p+v=q$ . This vector is denoted $q-p$ . Equivalently, the additive group of $V$ has a free and transitive group action on $A$ . (people.math.harvard.edu)

The vector space $V$ is called the translation space or direction space. Point differences obey

(q-p)+(r-q)=r-p.

Choosing a point $o$ identifies $A$ with $V$ through $p\mapsto p-o$ , but a different choice gives a different identification. Thus an affine space can acquire a vector-space structure after an origin is chosen, although that origin is not part of its original structure. (people.math.harvard.edu)

Coordinates, dimension, and frames

For finite-dimensional $V$ , the dimension of $A$ is $\dim_K V$ . An origin $o$ , together with a basis $e_1,\ldots,e_n$ , gives unique coordinates through

p=o+x_1e_1+\cdots+x_ne_n.

Changing the origin and basis changes coordinates by $x'=Mx+b$ , with $M$ invertible. Unlike a purely linear coordinate change, this includes a translation term. (cis.upenn.edu)

Points $p_0,\ldots,p_m$ are affinely independent when the vectors $p_1-p_0,\ldots,p_m-p_0$ are linearly independent. In dimension $n$ , an ordered collection of $n+1$ affinely independent points is an affine frame. Every point then has unique barycentric coordinates relative to that frame: coefficients whose sum is one. Two distinct points form a frame for a line; three noncollinear points form one for a plane. (cis.upenn.edu)

Affine combinations and convexity

The affine counterpart of a linear combination is an affine combination

\lambda_1p_1+\cdots+\lambda_mp_m, \qquad \sum_{i=1}^{m}\lambda_i=1.

Its intrinsic meaning is

o+\sum_{i=1}^{m}\lambda_i(p_i-o).

The condition on the coefficients makes this expression independent of $o$ : shifting the auxiliary origin changes the displacement sum by exactly the compensating amount. The notation therefore does not imply unrestricted addition of points. (cis.upenn.edu)

For distinct points $p,q$ , the points $p+t(q-p)$ , with $t\in K$ , constitute their line. Over the real numbers, restricting $t$ to $[0,1]$ gives the segment between them. More generally, an affine combination with nonnegative coefficients is a convex combination. A convex set contains all such combinations of its points. Affine sets contain every affine combination, including combinations with negative coefficients, so every real affine set is convex, but the converse fails. (kdd.cs.ksu.edu)

Affine subspaces and linear equations

A nonempty affine subspace has the form

B=p+W=\{p+w:w\in W\},

where $W$ is a linear subspace of the translation space. Its direction space is $W$ , and its dimension is $\dim W$ . Its affine structure does not depend on which point of $B$ is used as $p$ . In a vector space, it is a linear subspace precisely when it contains the zero vector. (cis.upenn.edu)

A consistent system of linear equations $Mx=b$ gives a central example. If $x_0$ is one solution, the entire solution set is

x_0+\ker M,

where $\ker M$ is the null space of the matrix $M$ . Subtracting two solutions gives a null-space vector; adding any null-space vector to a solution gives another solution. An inconsistent system has an empty solution set, not a nonempty affine space. A single nontrivial linear equation defines an affine hyperplane. (jhc.sjtu.edu.cn)

Affine maps and homogeneous coordinates

An affine map $f:A\to B$ has an associated linear map $L$ between direction spaces, satisfying

f(p+v)=f(p)+L(v).

It preserves affine combinations. In coordinates, its form is $f(x)=Mx+b$ ; it is bijective exactly when its linear part is invertible. A general affine map can collapse a line to a point, whereas a bijective affine map preserves lines, incidence, and parallelism. (wwwevs.mathematik.tu-darmstadt.de)

Using homogeneous coordinates, the coordinate column $x$ is extended to $(x,1)$ , and the map becomes

\begin{pmatrix}x\\1\end{pmatrix} \longmapsto \begin{pmatrix}M&b\\0&1\end{pmatrix} \begin{pmatrix}x\\1\end{pmatrix}.

Displacements are represented by $(v,0)$ , so the translation term affects points but not vectors. Successive affine transformations can consequently be composed using ordinary matrix multiplication, a representation used in computer graphics. (dgp.toronto.edu)

Relation to Euclidean geometry

An affine structure alone supplies neither lengths nor angles. Equipping its real direction space with a positive-definite inner product adds Euclidean measurements. General invertible affine maps may stretch or shear figures, so they need not preserve these measurements. Nevertheless, constructions defined by affine combinations remain meaningful: the centroid of a triangle is the combination of its three vertices with coefficients $1/3$ , independently of the coordinate origin. (maths.tcd.ie)