Affine Combinations

Most of linear algebra treats a subspace as the primary object: a set closed under all linear combinations, always containing the origin. Geometry needs lines and planes that miss the origin, triangles, and solid bodies. The device that produces them is a single restriction on the weights of a linear combination.¹

Given points ${\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_p$ in ${\mathbb{R}}^n$ and scalars $c_1,\dots ,c_p$, an affine combination is a linear combination

The set of all affine combinations of points in a set $S$ is the affine hull (or affine span) of $S$, written $\mathrm{aff}S$. The affine hull of a single point ${\mathbf{v}}_1$ is $\{{\mathbf{v}}_1\}$, since the only admissible weight is $c_1=1$.

The line through two points

For two distinct points ${\mathbf{v}}_1,{\mathbf{v}}_2$, an affine combination has weights $c_1+c_2=1$. Write $t=c_2$, so $c_1=1-t$, and

This passes through ${\mathbf{v}}_1$ at $t=0$ and ${\mathbf{v}}_2$ at $t=1$. Regrouping exposes its structure as a translate:

where $\mathbf{p}={\mathbf{v}}_1$ and $\mathbf{u}={\mathbf{v}}_2-{\mathbf{v}}_1$. The multiples $t\mathbf{u}$ trace $\mathrm{Span}\{\mathbf{u}\}$, the line through the origin in the direction $\mathbf{u}$. Adding $\mathbf{p}$ shifts that line to pass through ${\mathbf{v}}_1$. So $\mathrm{aff}\{{\mathbf{v}}_1,{\mathbf{v}}_2\}$ is the line through ${\mathbf{v}}_1$ and ${\mathbf{v}}_2$.

The relation between $\mathbf{y}$ and the shifted point $\mathbf{y}-{\mathbf{v}}_1$ holds for any number of points.

If $\mathbf{y}-{\mathbf{v}}_1=c_2({\mathbf{v}}_2-{\mathbf{v}}_1)+\cdots +c_p({\mathbf{v}}_p-{\mathbf{v}}_1)$, then collecting the ${\mathbf{v}}_1$ terms gives

whose weights sum to $1$. The converse reverses the algebra: $c_1=1-c_2-\cdots -c_p$ turns any affine combination back into a linear combination of the translated points. The point ${\mathbf{v}}_1$ carries no special role; any point in the list can play the base.

The translated-points criterion turns the membership question into a row reduction. Subtract ${\mathbf{v}}_1$ from every point and test whether $\mathbf{y}-{\mathbf{v}}_1$ lies in the span of the translated set.

When the points ${\mathbf{v}}_i$ are a basis $\mathcal{B}=\{{\mathbf{b}}_1,\dots ,{\mathbf{b}}_n\}$ for ${\mathbb{R}}^n$, the test is immediate: every $\mathbf{y}$ has unique $\mathcal{B}$-coordinates, and $\mathbf{y}$ is an affine combination of the basis exactly when those coordinates sum to $1$.

The affine hull as a flat

Although every linear combination of ${\mathbf{b}}_1,{\mathbf{b}}_2,{\mathbf{b}}_3$ fills all of ${\mathbb{R}}^3$, the affine combinations fill only the plane through the three points. That plane contains ${\mathbf{p}}_2$ and misses ${\mathbf{p}}_1$.

A set whose affine hull equals itself deserves a name. A set $S$ is affine if for every $\mathbf{p},\mathbf{q}\in S$ and every real $t$, the point $(1-t)\mathbf{p}+t\mathbf{q}$ is in $S$. Geometrically, whenever $S$ contains two points it contains the entire line through them.

The line condition covers combinations of two points by definition. An induction extends it to any number: for a combination $c_1{\mathbf{v}}_1+\cdots +c_{k+1}{\mathbf{v}}_{k+1}$ of points in $S$, at least one weight differs from $1$, say $c_{k+1}$. Set $t=c_1+\cdots +c_k=1-c_{k+1}\ne 0$. Then

expresses $\mathbf{y}$ as an affine combination of two points of $S$: the inner combination has weights summing to $1$ and lies in $S$ by the induction hypothesis.

Flats and their dimension

The definition ties affine sets to subspaces. A translate of a set $S$ by a vector $\mathbf{p}$ is $S+\mathbf{p}=\{\mathbf{s}+\mathbf{p}:\mathbf{s}\in S\}$. A flat in ${\mathbb{R}}^n$ is a translate of a subspace. Two flats are parallel if one is a translate of the other, and the dimension of a flat is the dimension of that parallel subspace.

• Line: a flat of dimension $1$.
• Hyperplane: a flat of dimension $n-1$.
• Point: a flat of dimension $0$.

In ${\mathbb{R}}^3$ the proper flats are points, lines, and planes, each of which may or may not pass through the origin. The dimension of an arbitrary set $S$, written $\dim S$, is the dimension of the smallest flat that contains it.

Fix $\mathbf{p}\in S$ and let $W=S-\mathbf{p}$. Since $\mathbf{p}\in S$, the zero vector lies in $W$. For ${\mathbf{u}}_1={\mathbf{s}}_1-\mathbf{p}$ and ${\mathbf{u}}_2={\mathbf{s}}_2-\mathbf{p}$ in $W$, a direct computation gives

The point ${\mathbf{s}}_1+{\mathbf{s}}_2-\mathbf{p}$ is an affine combination of points of $S$, hence in $S$, and one more affine combination keeps the result in $S$, so ${\mathbf{u}}_1+t{\mathbf{u}}_2\in W$. Thus $W$ is a subspace and $S=W+\mathbf{p}$ is a flat. The converse runs the same identity backward. The affine-sets-are-flats theorem gives the affine hull its geometric reading: $\mathrm{aff}S$ is the smallest flat containing $S$.

A familiar object is now recognizable as a flat.

Homogeneous coordinates

Affine combinations become ordinary linear combinations after one lift. For $\mathbf{v}$ in ${\mathbb{R}}^n$, the standard homogeneous form is

the point $\mathbf{v}$ lifted to the plane where the last coordinate equals $1$.

The last coordinate of $c_1{\tilde{\mathbf{v}}}_1+\cdots +c_p{\tilde{\mathbf{v}}}_p$ equals $c_1+\cdots +c_p$. Matching it to the last coordinate of $\tilde{\mathbf{y}}$, which is $1$, forces the weights to sum to $1$, and the first $n$ coordinates reproduce $\mathbf{y}=\sum c_i{\mathbf{v}}_i$. The affine constraint $\sum c_i=1$ has become the single linear equation last coordinate equals $1$.

Homogeneous coordinates carry into computer graphics, where translations, rotations, and perspective all become single matrix multiplications in ${\mathbb{R}}^{n+1}$. The affine structure of a scene reduces to linear algebra one dimension up, and the affine hull of a point set is the span of its lifted copies intersected with the height-$1$ plane. Affine independence — when those lifted copies are linearly independent — is the subject of barycentric coordinates.