Affine hull¶

In mathematics, the affine hull or affine span of a set S in Euclidean space \mathbb{R}^n is the smallest affine set containing S , or equivalently, the intersection of all affine sets containing S .

Version: v1 · 2026-09-28 · History
Domain-specific #: 7898
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Linear Algebra, Convex Geometry → Mathematics

Core Idea¶

Affine hull is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: In mathematics, the affine hull or affine span of a set S in Euclidean space \mathbb{R}^n is the smallest affine set containing S , or equivalently, the intersection of all affine sets containing S . In mathematics, the affine hull or affine span of a set S in Euclidean space \mathbb{R}^n is the smallest affine set containing S , or equivalently, the intersection of all affine sets containing S .

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The Endless Flat Through the Dots

Put two dots on a paper. The affine hull is the straight line that goes through both dots and keeps going forever in both directions. With three dots not in a line, it becomes a whole flat sheet that goes on forever. It is the smallest flat thing that holds all your dots.

Smallest Flat Through the Points

The affine hull of a group of points is the smallest perfectly flat shape, extending forever, that contains all of them. For two different points, it's the whole infinite straight line through them, not just the segment between them. For three points not on one line, it's the whole infinite flat plane through them. For one point, it's just that point. It's like asking, "What's the smallest flat world these points all live in?"

Smallest Affine Set Containing S

The affine hull of a set of points S is the smallest affine set that contains all of them, where an affine set is a flat like a point, line, or plane that may be shifted away from the origin, formally a translated vector subspace. You can build it from affine combinations: sums a1 x1 + ... + ak xk where the xi are points in S and the weights ai add up to 1 (weights can be negative, unlike in the convex hull, which only uses nonnegative weights). So two points give a whole infinite line, not just the segment between them. One point gives just that point. Another way to describe it: it is what the span of S would be if you moved the origin onto S.

In Euclidean space R^n, the affine hull (or affine span) of a set S, written aff(S), is the smallest affine set containing S, equivalently the intersection of all affine sets containing S. An affine set is a translate of a vector subspace. Concretely, aff(S) is the set of all affine combinations of elements of S: finite sums of alpha_i x_i with x_i in S, real coefficients alpha_i, and the coefficients summing to 1. Unlike the convex hull, the coefficients may be negative, so the affine hull of two distinct points is the full line through them rather than the segment. The affine hull of S is what span(S) would be if the origin were moved to a point of S; equivalently, aff(S) = x0 + span(S - x0) for any x0 in S. The affine hull of a singleton is the singleton itself.

Scope of Application¶

Examples. The affine hull of a singleton (a set made of one single element) is the singleton itself.
Examples. The affine hull of a set of two different points is the line through them.
Examples. The affine hull of a set of three points not on one line is the plane going through them.
Examples. The affine hull of a set of four points not in a plane in \mathbb{R}^3 is the entire space \mathbb{R}^3 .
Properties. \operatorname{aff}(\operatorname{aff} S) = \operatorname{aff} S \subset \operatorname{span} S = \operatorname{span} \operatorname{aff} S .

Clarity¶

A clear use of Affine hull names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics, the affine hull or affine span of a set S in Euclidean space \mathbb{R}^n is the smallest affine set containing S , or equivalently, the intersection of all affine sets containing S .

Manages Complexity¶

Affine hull compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—the affine hull of a set of three points not on one line is the plane going through them.—and the practical consequence—\operatorname{aff}(\operatorname{aff} S) = \operatorname{aff} S \subset \operatorname{span} S = \operatorname{span} \operatorname{aff} S .

Abstract Reasoning¶

Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
State the relation. Use the source-grounded identity: In mathematics, the affine hull or affine span of a set S in Euclidean space \mathbb{R}^n is the smallest affine set containing S , or equivalently, the intersection of all affine sets containing S .
Check operation and conditions.

Knowledge Transfer¶

Within the home domain. Knowledge about Affine hull transfers literally when a new case preserves the same carrier type, relation, and recognition test. The affine hull of a singleton (a set made of one single element) is the singleton itself. The affine hull of a set of two different points is the line through them. Beyond the home domain. No canonical parent is asserted for Affine hull. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Relationships to Other Abstractions¶

Current abstraction Affine hull Domain-specific

Parents (1) — more general patterns this builds on

Affine hull is a decomposition of Span Prime

Affine hull is the affine-geometric form of generating the smallest closed carrier containing declared generators.

Hierarchy paths (2) — routes to 2 parentless roots

Affine hull → Span → Closure

Show alternative path (1)

Neighborhood in Abstraction Space¶

Affine hull sits in a moderately populated region (52^nd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Geometric Figures & Constructions (32 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08