15  Affine Spaces

\(\mathbb{R}^n\) gets used for two different jobs that quietly get treated as one. Sometimes its elements are points — locations, with no privileged “zero location.” Sometimes they’re vectors — displacements, which very much have a privileged zero (no movement at all). This chapter names the structure that keeps the two apart, and the reason it’s worth doing: points don’t add, but the difference of two points is always a vector.

15.1 The problem, before the definition

Consider the set of points on a page. Nothing about the page hands you a natural origin — “the zero point” is a choice you impose, not a fact about the page. And yet you can still say something precise and origin-independent: “walk from point \(A\) to point \(B\)” describes a definite displacement, a vector, even before any origin has been chosen. Translate every point on the page 3 units right, and that translation is described by one vector, regardless of which point you use to anchor the description. Vector Spaces’s \(\vec{0}\) has no counterpart among points — there is no “zero point,” only displacements between points.

15.2 The abstraction

An affine space is a set \(A\) (its elements called points), a vector space \(V\) (its elements called displacements), and an action \(+: A \times V \to A\) satisfying:

  1. \(a + \vec{0} = a\) for every point \(a\).
  2. \((a + v) + w = a + (v+w)\) for all points \(a\) and displacements \(v,w\).
  3. For any two points \(a,b \in A\), there is a unique displacement \(v\) with \(a + v = b\) — written \(v = b - a\).

15.3 Why these axioms

The first two say translating is compatible with \(V\)’s own structure: translating by nothing changes nothing, and translating twice composes exactly the way adding the two displacements would. Without them, “displacement” wouldn’t behave consistently with the vector space it’s supposed to be drawn from. The third is the axiom doing the real work: it guarantees that “the displacement from \(A\) to \(B\)” is always a well-defined, unique vector — which is precisely the formal content of “you can subtract two points to get a vector, but you cannot add two points.” Without uniqueness, \(b - a\) could be ambiguous, and the entire idea of “the displacement between two specific points” would fail to pick out one answer.

15.4 What follows

An affine combination of points \(a_1,\dots,a_k\) with weights \(c_1,\dots,c_k\) summing to exactly \(1\) — \(c_1a_1 + \cdots + c_ka_k\), read as “start somewhere and combine displacements” — is a well-defined point, even though \(a_1 + a_2\) alone is not. The midpoint of two points, \(\frac{1}{2}a + \frac{1}{2}b\), is the simplest example, and the “weights sum to 1” requirement is not a technicality: it’s exactly what makes the result a point rather than a displacement (an arbitrary weighted sum of points with weights summing to something else isn’t meaningful in \(A\) at all). Anyone who has used barycentric coordinates or linear interpolation between positions has relied on this constraint without necessarily having seen why it’s required.

An affine map \(f: A \to A'\) has the form \(f(a) = a_0' + T(a - a_0)\) for a linear map \(T\) and fixed points \(a_0, a_0'\) — informally, “a linear map, plus a translation.” This is the direct generalization of Linear Algebra’s linear maps to a setting with no origin: recall that a linear map is required to satisfy \(T(\vec{0}) = \vec{0}\), fixing the origin — an affine map drops exactly that requirement, since an affine space has no origin to fix in the first place.

15.5 What an affine space is NOT

Not a vector space — it has no distinguished zero point, and points cannot be added to each other, only translated by displacements. \(\mathbb{R}^n\) can be viewed as an affine space over itself once an origin is chosen, and that choice is exactly what makes the point/vector distinction invisible in ordinary computation: picking an origin silently identifies “the point \((3,4)\)” with “the displacement \((3,4)\) from that origin,” and it’s easy to stop noticing they were ever conceptually different roles.

15.6 Where this leads

Euclidean Geometry, next, combines this chapter’s points-and-displacements structure with Inner Product Spaces’s length-and-angle structure — an affine space whose displacement vector space carries an inner product is, formally, what “Euclidean space” now means.