Affine combination · aiwiki.page

An affine combination is a linear combination of finitely many points or vectors in which the scalar coefficients sum to one. Unlike a convex combination, it permits negative coefficients. Affine combinations describe points on lines, planes, and higher-dimensional affine subspaces, and provide the algebraic foundation for barycentric coordinates. (stanford.edu)

Definition

For points $p_1,\ldots,p_m$ in a vector space over a field $K$ , an affine combination has the form

p=\sum_{i=1}^{m}\lambda_i p_i, \qquad \sum_{i=1}^{m}\lambda_i=1, \qquad \lambda_i\in K.

For geometric applications, the field is usually the real numbers. The defining restriction concerns only the sum of the coefficients: individual coefficients may be negative, zero, or greater than one. (stanford.edu)

The same operation is meaningful in an abstract affine space, where points do not have a distinguished zero and cannot ordinarily be added or multiplied by scalars. If $V$ is its associated vector space and $o$ is any reference point, define

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

Here each difference $p_i-o$ is a vector, and adding the resulting vector to $o$ produces a point. This is the intrinsic interpretation of the notation $\sum_i\lambda_i p_i$ . (cis.upenn.edu)

Why the coefficients sum to one

The sum-one condition makes the construction independent of the reference point. To see this directly, replace $o$ by $o'=o+v$ . The resulting point is

\begin{aligned} o'+\sum_i\lambda_i(p_i-o') &=o+v+\sum_i\lambda_i\bigl((p_i-o)-v\bigr)\\ &=o+\sum_i\lambda_i(p_i-o) +\left(1-\sum_i\lambda_i\right)v. \end{aligned}

The last term vanishes when $\sum_i\lambda_i=1$ . Thus an affine combination depends on the input points and their coefficients, not on an arbitrary choice of origin. This calculation explains why affine geometry distinguishes combinations of points from unrestricted combinations of vectors. (cis.upenn.edu)

A complementary identity is

\sum_i\mu_i p_i =\sum_i\mu_i(p_i-o) \quad\text{when}\quad \sum_i\mu_i=0,

where the right-hand side is a vector. Sum-one coefficients therefore describe a point, while sum-zero coefficients describe a displacement vector, without requiring a preferred origin. (cis.upenn.edu)

Geometric interpretation

For two distinct points $a$ and $b$ , every affine combination can be written as

p(t)=(1-t)a+tb=a+t(b-a).

Over the real numbers, varying $t$ over all of $\mathbb R$ traces the entire line through the points. When $0\leq t\leq1$ , the point lies on their line segment; $t<0$ or $t>1$ gives extrapolation beyond an endpoint. (stanford.edu)

For example, if $a=(0,0)$ and $b=(2,0)$ , then

\tfrac12a+\tfrac12b=(1,0), \qquad -a+2b=(4,0).

Both expressions are affine combinations, but only the first is convex.

The affine combinations of three noncollinear points fill their plane. Restricting the coefficients to be nonnegative instead gives the filled triangle, including its boundary. Four noncoplanar points similarly generate three-dimensional affine space, while their convex combinations form a tetrahedron. (cis.upenn.edu)

Affine hull and affine independence

The affine hull of a nonempty set $S$ , written $\operatorname{aff}(S)$ , is the set of all finite affine combinations of its points. It is the smallest affine set containing $S$ . In a real vector space, affine sets are precisely the sets closed under all such combinations. (stanford.edu)

For a finite collection, eliminating the first coefficient gives

\operatorname{aff}\{p_0,\ldots,p_k\} = p_0+\operatorname{span}\{p_1-p_0,\ldots,p_k-p_0\}.

This formula relates affine hulls to linear spans: the affine hull is a translated linear subspace. (cis.upenn.edu)

Points $p_0,\ldots,p_k$ are affinely independent when the difference vectors

p_1-p_0,\ldots,p_k-p_0

are linearly independent. Each point of their affine hull then has a unique affine representation. Its coefficients are called barycentric coordinates relative to the given points. If the points are affinely dependent, the representation need not be unique. (home.zcu.cz)

For example, a point in the plane of a nondegenerate triangle has three barycentric coordinates whose sum is one. It belongs to the triangle exactly when all three coordinates are nonnegative. (cis.upenn.edu)

Preservation by affine maps

An affine map preserves affine combinations. In vector-space coordinates, write it as

F(x)=Lx+b,

where $L$ is a linear map and $b$ is fixed. The preservation identity follows directly:

\begin{aligned} F\left(\sum_i\lambda_i p_i\right) &=L\left(\sum_i\lambda_i p_i\right)+b\\ &=\sum_i\lambda_i(Lp_i+b) =\sum_i\lambda_iF(p_i), \end{aligned}

because $\sum_i\lambda_i=1$ . Consequently, constructing an affine combination before applying an affine transformation gives the same result as transforming the input points first. (cis.upenn.edu)

Computing the coefficients

For specified points $p_1,\ldots,p_m\in\mathbb R^n$ and a target point $x$ , the defining equations can be assembled into a linear system:

\begin{pmatrix} | & & |\\ p_1 & \cdots & p_m\\ | & & |\\ 1 & \cdots & 1 \end{pmatrix} \begin{pmatrix} \lambda_1\\ \vdots\\ \lambda_m \end{pmatrix} = \begin{pmatrix} x\\ 1 \end{pmatrix}.

This is simply the coordinate equation $x=\sum_i\lambda_i p_i$ together with the sum-one constraint. A solution exists exactly when $x$ lies in the affine hull; it is unique exactly when the input points are affinely independent. The final row also connects affine combinations to the representation of points by homogeneous coordinates. (cis.upenn.edu)

Relation to convex combinations

Every convex combination is affine, but the converse is false. The distinction separates two different generated sets:

Affine combinations generate the affine hull.
Nonnegative affine combinations generate the convex hull.

The convex hull is contained in the affine hull and is a convex set. Negative coefficients are essential when representing points beyond the convex hull, although their presence alone does not guarantee that the resulting point lies outside it when the input points are affinely dependent. This last qualification follows from the possible nonuniqueness of affine representations. (stanford.edu)

The sum-one condition is therefore a condition of origin independence, not a condition of positivity or boundedness.

References

Convex Optimizationstanford.edu
Chapter 2: Basics of Affine Geometrycis.upenn.edu
Affine Combinations of Points and Barycentric Coordinateshome.zcu.cz