A representation of a point as a weighted affine combination of the vertices of a simplex, given by weights that sum to one; coordinates indicate relative contribution of each vertex.

Principle

Affine decomposition: any point in the affine span of an affinely independent vertex set can be written uniquely as a convex (or affine) combination of those vertices when the set is a simplex.

Demonstration

For a triangle in the plane, a point inside can be expressed by three nonnegative weights summing to one; those weights correspond to signed areas of subtriangles divided by the area of the whole triangle.

Misapplication

Using barycentric coordinates on a degenerate simplex (collinear vertices) and assuming uniqueness, or restricting to nonnegative weights when the point lies outside the simplex without acknowledging signed weights.

Consequence

Affine-invariant interpolation and easy formulation of finite-element shape functions and geometric algorithms; barycentric weights transform covariantly under affine maps.

Reversal

Cartesian coordinates expressed relative to an origin and basis rather than as weights on simplex vertices — they give absolute components rather than affine convex coefficients.

Boundary

Defined with respect to a chosen simplex (affinely independent vertex set); uniqueness fails if vertices are affinely dependent, and barycentric formulas require working in the simplex's affine span.

Semantic Tension

Homogeneous coordinates versus barycentric coordinates: homogeneous coordinates represent projective points using proportional tuples, while barycentric coordinates represent affine combinations with the sum-to-one normalization.

Synthesis

Barycentric coordinates assign to each point in the affine span of a simplex a tuple of weights summing to one that reconstruct the point as the weighted average of the simplex vertices, facilitating affine-invariant interpolation.