affine plane

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Definition:

  • An affine plane is a set of points and lines that satisfies the following three axioms:
    • A1: Each pair of distinct points determines a line.
    • A2: Parallel postulate: Given a point p and a line l, there exists exactly one line through p parallel to l.
    • A3: There exists a set of four points such that no line contains more than two of them.
  • Compare these to the axioms for a projective plane:
    • A1: Each pair of distinct points determines a line.
    • A2: Each pair of distinct lines intersects in a point.
    • A3: There exists a set of four points such that no line contains more than two of them.
  • Notice that the difference is in A2. The distinction is this: in an affine plane, there are parallel lines. In fact, given any line, as long as there's another point not on that line, there's a parallel to it. In a projective plane, there are NO parallel lines at all - all lines must intersect.

Cautions:

  • As with the projective plane, "point" and "line" - they are undefined terms, and lines don't have to be straight.
  • "parallel" is going to look a little odd as well - the key feature of parallel lines here is that they do not intersect.

Interpreting the axioms:

  • A1 and A3 function exactly as in the projective plane (so read that link if you haven't already).
  • A2 tells us that every line will have a parallel (remember, A3 rules out the possibility of all points co-linear, so if there's a line, there's automatically a point not on it through which a parallel will be drawn).

The smallest affine plane:

  • The smallest affine plane is the 4-Point Affine Plane. Here's a model:
  • affine plane sets
    affine plane
  • To visually represent the relationship between the sets, one of the "lines",{2,4}, appears curved outside the square - we do not want to draw it as a diagonal and give the impression that it intersects with {1,3} when it doesn't.

Exercise:

  • Verify for yourself that the 4-point affine plane satisfies the three axioms - consider all possible intersections of pairs of lines and observe that the intersection is either one point or an empty set (A1). The set of four points with the property that no line contains more than two of them(A3) is trivial - there are only four points to begin with. To verify A2, verify that for every possible combination of line + point not on line, there exists a parallel through the point (for example, for the line {1,2} and the point 3, the line through 3 parallel to {1,2} is {3,4}).

 

Other affine planes:

  • An affine plane of order n will have n2 points and n2 + n lines. Each line contains n points, and each point belongs to n + 1 lines. The 4-point plane is order 2, with 4 points, and 6 lines. When n = 3, we get the 9-point plane, which should have 12 lines.

And the obligatory Wiki reference:

http://en.wikipedia.org/wiki/Finite_geometry