Geometry self-study notes
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Starting with Michèle Audin's book Geometry.
1. Affine spaces
1.1. Definition
An affine space is:
- a set
- with the data of a vector space
(over a field of characteristic ) - and a mapping
. - Furthermore, if
, then we need the partial map to be a bijection from to . - Also (Chasles' relation),
. This is essentially the law of vector addition.
Dimension of
Instance: Every vector space is canonically an affine space by
choosing
Chasles' relation implies
An affine subspace is a subset of an affine space
Point
An affine map is a map
The affine bijections from
1.2. Facts
Any affine subspace of
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
If
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
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