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Starting with Michèle Audin's book Geometry.

1 Affine spaces

1.1 Definition

An affine space is:

  • a set \(\mathcal{E}\)
  • with the data of a vector space \(E\)(over a field of characteristic \(0\))
  • and a mapping \(\Theta : \mathcal{E}\times\mathcal{E} \longrightarrow E\).
  • Furthermore, if \(\Theta: (A, B)\mapsto\overrightarrow{AB}\), then we need the partial map \(\Theta_A : B \mapsto \overrightarrow{AB}\) to be a bijection from \(\mathcal{E}\) to \(E\).
  • Also (Chasles' relation), \(\forall A, B, C \in\mathcal{E}, \overrightarrow{AB} = \overrightarrow{AC} + \overrightarrow{CB}\). This is essentially the law of vector addition.

\(\Theta\) identifies vectors in \(E\) with orderd pairs of points in the affine space \(\mathcal{E}\).

Dimension of \(\mathcal{E}\) is the dimension of \(E\).

Instance: Every vector space is canonically an affine space by choosing \(\Theta: (u, v) \mapsto v - u\). Every affine space can be made a vector space by picking an origin point, but this is non-canonical since it depends on the choice of point.

Chasles' relation implies \(\overrightarrow{AA} = 0\) and \(\overrightarrow{AB} = -\overrightarrow{BA}\). It also implies that parallelogram rule of vector addition: \[\overrightarrow{AB} = \overrightarrow{A'B'} \iff \overrightarrow{AA'} = \overrightarrow{BB'}.\]

An affine subspace is a subset of an affine space \(\mathcal{E}\) such that image of subset under partial map \(\Theta_A\) (for some \(A\in\mathcal{E}\)) is a vector subspace of \(E\). Then we can define points, lines and planes to just be affine subspaces of dimension 0, 1, and 2 respectively.

Point \(A_0, A_1, \ldots, A_k\) are affinely independent if the dimension of the affine space they span is \(k\). If \(k = dim(\mathcal{E})\), then it is said that \((A_0, \ldots, A_k)\) is an affine frame of \(\mathcal{E}\). For example, an affine frame on a line has two distinct points.

An affine map is a map \(\phi: \mathcal{E}_1 \rightarrow \mathcal{E}_2\) if there exists a point \(O\) (which we can call the origin) and a linear map \(f: E \rightarrow F\) such that \(\forall A\in\mathcal{E}\), \[f(\overrightarrow{OA}) = \overrightarrow{\phi(O)\phi(A)}.\]

The affine bijections from \(\mathcal{E}\) to itself form a group under composition called the affine group \(GA(\mathcal{E})\).

1.2 Facts

Any affine subspace of \(\mathcal{E}\) can be written as \(V + \overrightarrow{v}\), where \(V\) is a vector subspace of \(E\) and \(\overrightarrow{v}\in E\). The vector subspaces are precisely the affine subspaces with \(\overrightarrow{v}=0\). (See also analogous result for affine maps).

An intersection of affine subspaces is an affine subspace.

The underlying vector space gives the "direction" to an affine space. This lets us also talk about parallel affine spaces (they have the same direction).

If \(f : E \rightarrow V\) is a linear map then \(f^{-1}(v)\) for any \(v\in F\) is an affine subspace of the affine space \(E\) having \(Ker(f)\) as its underlying vector space. Moreover, \(f^{-1}(v)\) is parallel to every other \(f^{-1}(v')\).

If \(\mathcal{E}_1, \mathcal{E}_2\) are parallel, they are either equal or disjoint.

Surprisingly: the parallel axiom (there exists a line parallel to a given line through a point not on the line) is true in any affine space (because it is actually a consequence of vector space axioms and hence of the definition of parallelism).

As a result of this definition of affine maps, given an affine map, we can always define a canonical linear map. For example, the linear map associated to the affine map \(x\mapsto ax+b\) is \(x\mapsto ax\). Every affine map \(\phi\) can be written as \(\phi(v) = f(v) + v_0\) for some linear map \(f\) and some vector \(v_0\). (See analogous result for affine subspaces).

The forward as well as inverse image of an affine space by an affine map is an affine space. The composition of two affine mappings is an affine mapping.

Surprisingly: The fact that the three medians of a triangle meet at a single point is a consequence of the associativity of computing barycenters (centers of gravity) of a system of masses.

Any intersection of convex subsets is convex.

The forward and inverse image of a convex subset by an affine mapping is convex.

1.3 Theorems

1.3.1 Thales' theorem

This theorem states that projections are affine maps. Equivalently, any transversal cutting accross three fixed parallel lines will have "segements" cut by the parallel lines in a fixed ratio.

1.3.2 Others

Other notable (classical) geometry theorems include:

  • Pappus' theorem
  • Desargues' theorem
  • Menela\"{u}s' theorem
  • Ceva's theorem
  • (not so clasiscal) The fundamental theorem of affine geometry

2 TODO Euclidean geometry, generalities

Author: Vaibhav Karve

Created: 2021-04-08 Thu 10:49

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