Definition·D041
Affine plane
Points and lines obeying three laws: joining, parallels, non-degeneracy.
An affine plane is an ordered pair
, where
is a set of points and
is a set of lines, each line a subset of
, such that:
In words
An affine plane consists of points and lines such that: two distinct points lie on exactly one common line; through a point off a line passes exactly one line missing it; and not all points are collinear.
Rests onA02
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).
Remarks
Lines are literally sets of points (
sits inside the power set, via Power set and D001), so the whole geometry is an object of set theory and its theorems are ZFC theorems about every structure satisfying the three axioms. (AP2) is Playfair's axiom, the modern form of Euclid's fifth postulate: one parallel, no more, no less; its disjointness clause uses intersection and the empty set. (AP3) rules out degenerate cases (empty planes, single lines). The smallest model has
points and
lines, each line a two-element subset, like the vertices and edges of a tetrahedron; the classical model is the real coordinate plane, which awaits a construction of the reals. First consequences: two lines meet in at most one point (L31), and parallelism is an equivalence relation.