##  [Affine Space](/affine-space-0) 

 Definition

A set of points with a transitive free action of a vector space that models translation without a distinguished origin; vectors represent differences between points, and parallelism is defined while no unique zero point is chosen.

 

 

 

 

 

 





## Principle

Principle

An affine space A over a vector space V satisfies that for any p in A and v in V there is a point p+v in A, and for any two points p,q the difference q−p is a vector in V; affine combinations preserve sums of coefficients equal to one.

 

 

 

 

 





## Demonstration

Demonstration

The Euclidean plane regarded without a chosen origin is an affine plane: coordinates may be introduced by selecting a reference point and basis, but geometric statements about parallel lines or barycenters do not depend on that choice.

 

 

 

 

## Misapplication

Misapplication

Treating an affine space as a vector space by fixing an arbitrary origin and forgetting that such a choice is not canonical can obscure invariance properties and produce coordinate-dependent statements erroneously presented as intrinsic.

 

 

 

 

 





## Consequence

Consequence

Using affine structure focuses on invariant geometric concepts (collinearity, parallelism, barycenters) and underpins projective constructions and coordinate-free formulations in geometry and applied mathematics.

 

 

 

 

## Reversal

Reversal

The reversal is a vector space with a distinguished origin; adding a canonical zero and linear structure removes the origin‑free symmetry and changes which operations are intrinsic.

 

 

 

 

 





## Boundary

Boundary

Affine spaces presuppose an underlying vector space of translations but do not carry an inner product or metric unless extra structure is specified; they exclude quotient identifications that collapse parallelism (e.g., some projective quotients).

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension exists between 'affine' and 'vector' viewpoints: vectors encode differences, points lack origin; many authors blur the line by choosing coordinates, causing friction between coordinate-dependent and coordinate-free descriptions.

 

 

 

 

 





## Synthesis

Synthesis

An affine space is the origin-free geometric setting where points and translation vectors coexist: it preserves parallel and barycentric relations while deferring any choice of origin or metric, thus unifying coordinate-free geometry with classical constructions.