Affine Independence and Barycentric Coordinates

Linear dependence asks whether some nontrivial combination of vectors is zero. Affine dependence adds one requirement: the weights must also sum to zero. That extra equation is what separates the geometry of triangles and simplices from the algebra of subspaces.¹

An indexed set $\{{\mathbf{v}}_1,\dots ,{\mathbf{v}}_p\}$ in ${\mathbb{R}}^n$ is affinely dependent if there exist scalars $c_1,\dots ,c_p$, not all zero, with

Otherwise the set is affinely independent. The prototype is three points on a line. If ${\mathbf{v}}_3=(1-t){\mathbf{v}}_1+t{\mathbf{v}}_2$, then $(1-t){\mathbf{v}}_1+t{\mathbf{v}}_2-{\mathbf{v}}_3=0$, a linear dependence whose weights $(1-t)+t-1=0$ also sum to zero: an affine dependence.

Because the weights carry an extra constraint, affine dependence is stricter than linear dependence. Every affinely dependent set is linearly dependent, but not conversely.

The third condition is the test in practice: subtract one point from all the others and check the resulting $p-1$ vectors for linear dependence by row reduction. The fourth condition tests the lifted points directly, without choosing a base point.

When a fourth point lands on the plane, the set becomes affinely dependent, and the extra point acquires weights that record its position relative to the first three.

Barycentric coordinates

Affine independence gives each point of an affine hull a unique weight vector, the exact analogue of coordinates relative to a basis.

The weights $c_1,\dots ,c_k$ are the barycentric coordinates of $\mathbf{p}$. Uniqueness follows from affine independence: two representations would subtract to an affine dependence with weights summing to zero. The computation uses homogeneous forms, since the two conditions of the unique-representation theorem combine into

Row reduction of $[{\tilde{\mathbf{v}}}_1\cdots {\tilde{\mathbf{v}}}_k\mid \tilde{\mathbf{p}}]$ produces the coordinates.

Physical and geometric meaning

Barycentric coordinates were introduced by Möbius in 1827 for a point inside a triangle. Place masses $m_a,m_b,m_c$ at the vertices $\mathbf{a},\mathbf{b},\mathbf{c}$; the center of mass sits at $\mathbf{p}=m_a\mathbf{a}+m_b\mathbf{b}+m_c\mathbf{c}$ once the masses are scaled to sum to $1$. The coordinates are the normalized masses themselves, which is the origin of the term.

The geometric reading is a ratio of areas. Joining $\mathbf{p}$ to each vertex splits triangle $\mathbf{abc}$ into three smaller triangles, and

Each coordinate is the fraction of the total area cut off by the sub-triangle opposite that vertex.

When $\mathbf{p}$ leaves the triangle, some coordinates turn negative. The lines where a coordinate equals a fixed value are parallel to the opposite side: the line through $\mathbf{b}$ and $\mathbf{c}$ has $c_1=0$ (its points are affine combinations of $\mathbf{b}$ and $\mathbf{c}$ alone), and the parallel line through $\mathbf{a}$ has $c_1=1$. Crossing the side $\mathbf{bc}$ flips the sign of $c_1$.

Interpolation in computer graphics

A surface approximated by flat triangular patches carries data only at the vertices: a color, a normal direction, a depth. Barycentric coordinates spread that data across the interior. The value at an interior point $\mathbf{p}$ is the combination of the vertex values with the barycentric coordinates as weights — the interpolation that produces smooth shading from a coarse mesh.

Colors are stored as RGB triples $(r,g,b)$ with each component between $0$ and $1$. Interpolating a color across a triangle means applying the barycentric weights to the three vertex colors.

The same weights answer the point-in-triangle question. A point of the plane lies inside triangle $\mathbf{abc}$ precisely when all three barycentric coordinates are nonnegative; a zero coordinate puts it on an edge, and a negative one puts it outside. Ray tracing uses this: a ray $\mathbf{x}(t)=\mathbf{a}+t\mathbf{d}$ meets the plane of a triangle at one parameter $t$, and the sign pattern of the barycentric coordinates of that intersection decides whether the ray actually strikes the patch.

Nonnegative barycentric coordinates single out the points of the triangle from the points of its full plane. That restriction, applied to any set of points, is the definition of a convex combination.