Axioms
Affine space is usually studied as analytic geometry using coordinates, or equivalently vector spaces. It can also be studied as synthetic geometry by writing down axioms, though this approach is much less common. There are several different systems of axioms for affine space.
Coxeter (1969, p.192) axiomatizes affine geometry (over the reals) as ordered geometry together with an affine form of Desargues's theorem and an axiom stating that in a plane there is at most one line through a given point not meeting a given line.
Affine planes satisfy the following axioms (Cameron 1991, chapter 2): (in which two lines are called parallel if they are equal or disjoint):
- Any two distinct points lie on a unique line.
- Given a point and line there is a unique line which contains the point and is parallel to the line
- There exist three non-collinear points.
As well as affine planes over fields (or division rings), there are also many non-Desarguesian planes satisfying these axioms. (Cameron 1991, chapter 3) gives axioms for higher dimensional affine spaces.
Read more about this topic: Affine Space
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