Affine combination

An affine combination of points or vectors is a weighted sum whose coefficients sum to one. Unlike a convex combination, its coefficients may be negative.

Linear combination: A sum of vectors multiplied by scalar coefficients. An affine combination is a linear combination with the additional requirement that its coefficients sum to one.
Translation invariance: A property unchanged when every point is shifted by the same displacement. Unit-sum coefficients make a point's affine combination shift by exactly the common translation.
Linear interpolation: A method for estimating values between two known values using a weighted average. Interpolating between two points uses an affine combination with weights summing to one.
Conic combination: A linear combination whose coefficients are all nonnegative, with no requirement that they sum to one. It constrains coefficient signs but not their total, unlike an affine combination.
Affine space: A space of points in which differences are vectors, but no origin is distinguished. Affine combinations of points are intrinsic to affine spaces and do not depend on a choice of origin.
Affine independence: A set of points is affinely independent when no nonzero coefficients summing to zero give a vanishing weighted sum. The zero-sum relation tests uniqueness of affine coordinates among points.
Barycentric interpolation: Interpolation over a simplex using barycentric coordinates as weights. Its interpolated value is an affine combination of the values at simplex vertices.
Projective coordinates: Coordinates that represent points up to multiplication of all coordinates by the same nonzero scalar. They use scale-equivalent coordinates rather than the normalized unit-sum coefficients of affine coordinates.
Vector space: A set whose elements can be added and scaled while satisfying the vector-space axioms. Vector-space operations supply the addition and scalar multiplication used in affine combinations.
Affine hull: The smallest affine subspace containing a given set of points. It consists of all affine combinations of those points.

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