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Affine combination

In mathematics, an affine combination of points x_1, \dots, x_m in a vector space such as \mathbb{R}^n is defined as a sum \sum_{i=1}^m \lambda_i x_i, where the coefficients \lambda_i are scalars satisfying \sum_{i=1}^m \lambda_i = 1.^[1] This formulation ensures that the result remains invariant under translations, distinguishing it from general linear combinations where the coefficients need not sum to unity.^[2] Affine combinations form the basis for affine geometry, where they define affine subspaces as the smallest sets closed under such operations, encompassing lines, planes, and higher-dimensional flats translated from the origin.^[1] The affine hull of a set X \subseteq \mathbb{R}^n is precisely the collection of all affine combinations of elements from X, providing a way to characterize the affine span without relying on a fixed origin.^[1] A key property is that a set is affine if and only if it contains every affine combination of its points, enabling the study of geometric structures like barycenters, which represent weighted averages (allowing negative weights) of points.^[2]^[1] When the coefficients \lambda_i are restricted to be non-negative, an affine combination becomes a convex combination, which lies within the convex hull and is crucial for convex optimization and computational geometry.^[3] Affine independence of points x_0, \dots, x_m holds if no point is an affine combination of the others, analogous to linear independence for vectors, and ensures that their affine hull has dimension m.^[1] These concepts underpin affine transformations, which preserve affine combinations and are widely applied in computer graphics, robotics, and machine learning for modeling position-independent operations.^[2]

Fundamentals

Definition

In a vector space over a field, such as the real numbers \mathbb{R}, an affine combination of points x_1, x_2, \dots, x_n is given by the expression \sum_{i=1}^n \lambda_i x_i, where the scalars \lambda_i satisfy the condition \sum_{i=1}^n \lambda_i = 1.^[3] This construction requires the points to be elements of an affine space or vector space equipped with operations of addition and scalar multiplication.^[4] The coefficients \lambda_i are termed affine coefficients and are distinguished from the arbitrary weights used in general linear combinations by the normalization constraint that ensures their sum equals unity.^[5] A simple example arises with two points, where the affine combination \lambda x_1 + (1 - \lambda) x_2 represents a point on the line segment between x_1 and x_2 when $0 \leq \lambda \leq 1.^[3]

Historical Context

The term "affine" was introduced by Leonhard Euler in 1748. The concept of affine combinations traces its origins to 19th-century developments in geometry, particularly through the work of August Ferdinand Möbius and Hermann Grassmann. In his 1827 publication Der barycentrische Calcul, Möbius introduced barycentric coordinates as a method to represent points as weighted combinations of reference points, effectively describing what would later be recognized as affine combinations in the context of geometric relations without a fixed origin.^[6] Grassmann's 1844 work Die lineale Ausdehnungslehre developed foundational ideas for affine geometry in n-dimensional spaces. This approach predated modern vector spaces and emphasized affine invariants, such as ratios preserved under parallel projections, laying foundational ideas for affine geometry.^[7] The evolution continued with Felix Klein's Erlangen program in 1872, which provided a unifying framework for geometries based on transformation groups, with the affine group later recognized as a subgroup of the projective group in Klein's subsequent developments. Klein's Vergleichende Betrachtungen über neuere geometrische Forschungen classified various geometries as the study of properties invariant under specific transformation groups, shifting focus from Euclidean metrics to broader structural invariances and influencing the abstract treatment of combinations in geometric spaces.^[7] This program marked a transition toward more algebraic perspectives in the late 19th century, bridging earlier barycentric methods with emerging group-theoretic insights. In the early 20th century, David Hilbert advanced axiomatic approaches to geometry in his 1899 Grundlagen der Geometrie, which influenced later developments in affine and projective geometry.^[8] The explicit terminology "affine combination" emerged in linear algebra texts after the 1940s, influenced by the rise of abstract algebra and axiomatic approaches in works like those of the Bourbaki group, which formalized combinations in affine spaces as sums with coefficients totaling one to preserve affine structure.^[7]

Mathematical Properties

Key Properties

An affine combination of points x_1, \dots, x_n in \mathbb{R}^d is given by \sum_{i=1}^n \lambda_i x_i where the coefficients satisfy \sum_{i=1}^n \lambda_i = 1. The affine hull of a set of points \{x_1, \dots, x_n\} in \mathbb{R}^d is the set of all such affine combinations, denoted \{ \sum_{i=1}^n \lambda_i x_i \mid \sum_{i=1}^n \lambda_i = 1 \}, which forms the smallest affine subspace containing those points.^[9]^[10] The set of all affine combinations of a fixed set of points is closed under further affine combinations, meaning that if points are in the affine hull, any affine combination of them remains within that hull, thereby forming an affine subspace.^[9]^[11]^[10] A set of points \{x_1, \dots, x_n\} is affinely independent if none of the points can be expressed as an affine combination of the others, or equivalently, if the vectors \{x_2 - x_1, \dots, x_n - x_1\} are linearly independent; for an affinely independent set of n points, the dimension of the affine hull is n-1.^[9]^[11]^[10] For an affinely independent set of points, the representation of any point in their affine hull as an affine combination is unique, ensuring that the coefficients \lambda_i are determined solely by the positions of the points.^[9]^[11]^[10] Affine combinations are preserved under affine transformations, such as translations and linear maps, meaning that if f is an affine transformation, then f\left(\sum_{i=1}^n \lambda_i x_i\right) = \sum_{i=1}^n \lambda_i f(x_i) for \sum_{i=1}^n \lambda_i = 1.^[9]^[11]^[10]

Relation to Linear Combinations

An affine combination of points x_1, \dots, x_n in an affine space is defined as \sum_{i=1}^n \lambda_i x_i where the coefficients satisfy \sum_{i=1}^n \lambda_i = 1.^[9] In contrast, a linear combination \sum_{i=1}^n \lambda_i x_i imposes no such constraint on the coefficients \lambda_i, allowing their sum to be any real number.^[12] This restriction in affine combinations ensures that the result remains a point in the affine space, preserving the geometric structure under changes of origin, whereas unrestricted linear combinations can scale or shift the result in ways that depend on the chosen frame.^[2] Any affine combination can be equivalently expressed as a fixed point plus a linear combination of differences between the points, which are vectors. Specifically, \sum_{i=1}^n \lambda_i x_i = x_1 + \sum_{i=2}^n \lambda_i (x_i - x_1), where \lambda_1 = 1 - \sum_{i=2}^n \lambda_i.^[9] This rewriting highlights that the affine combination decomposes into a translation-invariant vector component (the linear combination of differences) anchored at a base point x_1.^[13] A key implication of this structure is that affine combinations are invariant under translations, as translations add the same vector to each point, preserving the condition \sum \lambda_i = 1 and the differences x_i - x_1.^[14] In general linear combinations, however, translations alter the result unless the coefficients sum to zero, emphasizing how affine combinations abstract away absolute position in favor of relative geometry.^[2]

Examples and Illustrations

Vector Space Examples

In one-dimensional vector spaces, such as the real line \mathbb{R}, affine combinations provide a straightforward way to locate points between given positions using weights that sum to 1. For instance, consider the points p_1 = 0 and p_2 = 3; the affine combination $0.7 \cdot p_1 + 0.3 \cdot p_2 = 0.7 \cdot 0 + 0.3 \cdot 3 = 0.9 yields the point 0.9, which lies between 0 and 3 weighted toward the origin. To illustrate negative weights, consider $1.5 \cdot p_1 + (-0.5) \cdot p_2 = 1.5 \cdot 0 + (-0.5) \cdot 3 = -1.5, which lies outside the interval [0, 3] on the line. This illustrates how affine combinations generalize interpolation on a line, preserving the affine structure where the result remains a point in \mathbb{R}.^[11]^[15] Extending to two dimensions in \mathbb{R}^2, affine combinations allow for positioning points within the plane formed by multiple vectors. Take the points p_1 = (0,0), p_2 = (1,0), and p_3 = (0,1); using weights 0.5, 0.3, and 0.2 respectively, the combination is

$0.5 \cdot (0,0) + 0.3 \cdot (1,0) + 0.2 \cdot (0,1) = (0,0) + (0.3,0) + (0,0.2) = (0.3, 0.2).

This point (0.3, 0.2) lies in the affine hull of the three points, demonstrating how non-equal weights distribute the position inside the triangle spanned by them.^[11]^[9] In higher dimensions, such as \mathbb{R}^3, equal-weight affine combinations compute centroids of simplices like tetrahedrons. For the vertices p_1 = (0,0,0), p_2 = (1,0,0), p_3 = (0,1,0), and p_4 = (0,0,1), the centroid is the affine combination with weights $1/4 each:

\frac{1}{4} \cdot (0,0,0) + \frac{1}{4} \cdot (1,0,0) + \frac{1}{4} \cdot (0,1,0) + \frac{1}{4} \cdot (0,0,1) = \left( \frac{1}{4}, \frac{1}{4}, \frac{1}{4} \right).

This barycenter represents the balanced center of mass assuming uniform density, a key application in geometric modeling. A non-trivial case arises with affine dependence, where points lie in a lower-dimensional affine subspace, allowing one to be expressed as an affine combination of the others. For three collinear points in \mathbb{R}^2, such as p_1 = (0,0), p_2 = (1,0), and p_3 = (2,0), the point p_2 is the affine combination $0.5 \cdot p_1 + 0.5 \cdot p_3 = 0.5 \cdot (0,0) + 0.5 \cdot (2,0) = (1,0), indicating affine dependence since the dimension of their affine hull is 1, less than 2.^[16]^[9] This dependence relation can also be seen through the nontrivial coefficients \alpha_1 = 1, \alpha_2 = -2, \alpha_3 = 1 satisfying \sum \alpha_i p_i = 0 and \sum \alpha_i = 0.^[17]

Geometric Interpretations

The set of all affine combinations of two distinct points in Euclidean space generates the affine line passing through them. The position along the line depends on the weights assigned to each point, provided the weights sum to one; for non-negative weights, the point lies on the line segment connecting them. Extending this, an affine combination of multiple points generates points within the affine subspace they span, such as a line for three collinear points or a plane for four coplanar points.^[9] Geometrically, an affine combination can be interpreted as the barycenter, or center of mass, of the points with masses proportional to the weights, normalized so the total mass is one; this ensures the resulting point is independent of the chosen origin in the space.^[9] In the plane, the set of affine combinations of the three vertices of a triangle, using non-negative weights that sum to one, fills the interior and boundary of the triangle itself.^[9] The affine span of k affinely independent points forms a (k-1)-dimensional flat, such as a line for k=2 or a plane for k=3, representing the lowest-dimensional affine subspace containing those points.^[18] Affine combinations preserve ratios of distances along lines under affine transformations; for instance, a point dividing a segment in a fixed proportion remains at that same proportion after the transformation.^[14]

Applications

In Affine Geometry

Affine spaces are geometric structures defined as sets of points that are closed under affine combinations, distinguishing them from vector spaces by lacking a distinguished origin. This closure ensures that any affine combination of points within the space remains a point in the space, providing a foundation for geometry without privileging a specific point as zero. Formally, given a set E over a field k, it forms an affine space if for any points a_i \in E and scalars \lambda_i \in k with \sum \lambda_i = 1, the combination \sum \lambda_i a_i \in E.^[19] In affine geometry, vector addition is constructed via the parallelogram law, where the sum of vectors \overrightarrow{ab} and \overrightarrow{ad} corresponds to the fourth vertex e of the parallelogram formed by the points a, b, d, expressible as an affine combination: specifically, the point e = a + (b - a) + (d - a) = b + d - a, with coefficients -1 for a, $1 for b, and $1 for d, summing to 1. Scalar multiplication similarly derives from affine combinations, such as scaling a vector by \lambda yielding \lambda \overrightarrow{ab} = a + \lambda (b - a), again closed under the affine structure. These operations underpin the translation-invariant properties of affine spaces, allowing geometric constructions without a fixed origin.^[10] Affine combinations play a central role in key theorems of affine geometry, defining parallelism and division ratios independently of metrics. For instance, two lines are parallel if and only if the affine combinations along them preserve ratios, as captured by Thales' theorem, which states that parallel hyperplanes divide segments in equal ratios via affine combinations. The midpoint theorem exemplifies this: in an affine triangle, the midsegments joining the midpoints of two sides—each defined as the affine combination \frac{1}{2}a + \frac{1}{2}b with equal weights—are parallel to the third side and half as long, forming the medial triangle. These theorems highlight how affine combinations enforce the parallel postulate and ratio preservation intrinsic to the geometry.^[9]^[10] A defining feature of affine geometry is that affine maps—transformations between affine spaces—preserve affine combinations, mapping \sum \lambda_i a_i to \sum \lambda_i f(a_i) for \sum \lambda_i = 1. This preservation classifies affine geometries up to isomorphism, as the fundamental theorem of affine geometry asserts that any bijective map preserving lines (or equivalently, affine combinations) is an affine transformation, composed of a linear map and translation.^[20] Such maps enable the study of invariants like parallelism and ratios, distinguishing affine from other geometries. Affine combinations also facilitate connections to projective geometry, where an affine space embeds as a subset excluding a hyperplane at infinity, with affine combinations restricted to finite points. For example, parallel lines in the affine plane intersect at a point at infinity in the projective closure, but affine subsets remain closed under combinations without invoking infinite elements.^[10]

In Convex Analysis

In convex analysis, the convex hull of a set S in a Euclidean space is defined as the smallest convex set containing S, and it coincides with the set of all convex combinations of points from S. A convex combination is a special case of an affine combination where the coefficients \lambda_i satisfy \lambda_i \geq 0 for all i and \sum \lambda_i = 1, ensuring that the resulting point lies within the "filled" region bounded by the points in S. This representation is crucial for understanding the structure of convex sets, as it allows any point in the convex hull to be expressed explicitly as such a weighted average. The Krein–Milman theorem extends this idea to infinite-dimensional settings, stating that a nonempty compact convex subset K of a locally convex Hausdorff topological vector space is equal to the closure of the convex hull of its extreme points. Extreme points of K are those elements that cannot be written as a nontrivial convex combination of distinct points in K, and the theorem implies that K can be generated via convex combinations (and limits thereof) solely from these boundary elements. This result underpins much of functional analysis and optimization by reducing the study of entire convex sets to their "corners." Affine combinations, particularly their convex variants, are central to optimization problems such as linear programming, where the feasible region is a convex polyhedron formed by the intersection of half-spaces defined by linear inequalities. Any feasible point in this region can be expressed as a convex combination of the polyhedron's vertices, which are its extreme points. Carathéodory's theorem refines this by showing that, in d-dimensional space, such a representation requires at most d+1 vertices that are affinely independent. For instance, in optimization, a point x in the feasible region satisfies

x = \sum_{i=1}^{k} \lambda_i v_i,

where k \leq d+1, each v_i is a vertex, \lambda_i \geq 0, and \sum_{i=1}^{k} \lambda_i = 1. This decomposition facilitates algorithms like the simplex method, which navigate vertices to find optima.^[21]

Extensions and Generalizations

Convex Combinations

A convex combination is a special case of an affine combination where all coefficients are non-negative.^[22] Specifically, given points x_1, x_2, \dots, x_k in a vector space, a point x is a convex combination of these points if there exist coefficients \lambda_1, \lambda_2, \dots, \lambda_k \geq 0 such that \sum_{i=1}^k \lambda_i = 1 and x = \sum_{i=1}^k \lambda_i x_i.^[23] This formulation distinguishes convex combinations from general affine combinations, where coefficients may be negative, allowing points outside the convex hull to be represented, whereas convex combinations are restricted to points within or on the boundary of the convex hull generated by the original points.^[24] The non-negativity ensures that convex combinations lie within the convex hull defined by the points, without extending beyond it via negative weights.^[25] The set of all convex combinations of a finite set of points forms the convex hull of that set, which is the smallest convex set containing the points.^[26] Moreover, the convex hull is closed under convex combinations: any convex combination of points within the hull remains in the hull, preserving the convexity property.^[27] In statistics, convex combinations appear as weighted averages of probability distributions, where the weights are probabilities summing to 1, such as in mixture models that blend multiple distributions non-negatively.^[28]

Barycentric Coordinates

Barycentric coordinates provide a coordinate system for points in an affine space relative to an affinely independent set of basis points. For an affinely independent set of points \{v_1, \dots, v_{n+1}\} in an n-dimensional affine space, any point p in the affine hull can be uniquely expressed as an affine combination

p = \sum_{i=1}^{n+1} \lambda_i v_i,

where the coefficients \lambda_1, \dots, \lambda_{n+1} satisfy \sum_{i=1}^{n+1} \lambda_i = 1. These coefficients (\lambda_1, \dots, \lambda_{n+1}) are the barycentric coordinates of p with respect to the basis.^[29] A key property of barycentric coordinates is that they always sum to 1, reflecting the affine nature of the combination. In the context of a simplex formed by the basis points, the barycentric coordinates of interior points are all positive, while points on the boundary have at least one zero coordinate. The simplex itself corresponds to the standard simplex in barycentric coordinates, where all \lambda_i \geq 0 and sum to 1. This representation generalizes naturally to higher dimensions, allowing points in n-simplices to be coordinatized similarly.^[29] In computational geometry, barycentric coordinates facilitate interpolation within simplices, such as triangles in 2D. For a point p inside triangle \triangle ABC, the coordinates (\lambda_A, \lambda_B, \lambda_C) serve as weights for interpolating attributes like color or texture from the vertices:

p = \lambda_A A + \lambda_B B + \lambda_C C, \quad \lambda_A + \lambda_B + \lambda_C = 1.

These weights are computed as ratios of signed areas of sub-triangles: \lambda_A = \frac{\area(\triangle pBC)}{\area(\triangle ABC)}, and similarly for the others, enabling efficient point location and shading in rendering algorithms.^[30] Barycentric coordinates extend to higher-dimensional applications, including finite element methods for solving partial differential equations on simplicial meshes. In these methods, they define shape functions or basis functions that are 1 at a vertex and 0 at others, ensuring interpolation consistency across elements in 3D tetrahedra or beyond.^[31]

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