Given an affine plane, we can introduce points at “infinity” and join those points by a line to get a system in which there are no parallel lines and any two distinct lines determine a unique point! More formally, what we do is this :
- Corresponding to each parallel class we take a single new point and add that point to all the lines of that class. Different parallel classes get different points and these points are called points at infinity.
- All the new points at infinity (and only these) form a single new line called the line at infinity.
Note here that infinity is just a terminology, origins of which will be clear if we study the history of projective geometry.
Def : A projective plane is a triple where is a set of points, a set of lines and an incidence relation such that :
- Each line is incident with at least three points and each point is incident with at least three lines.
- Any two distinct points determine (/ lie on) a unique line.
- Any two distinct lines determine (/ intersect at) a unique point.
Note that the definition of a projective plane has a duality between lines and points; if the words lines and points are exchanged in the definition, then all the axioms are preserved. Thus any proof about lines can be turned into a proof about points, and vice versa.
Theorem 1 Every affine plane can be extended to a projective plane and every projective plane has an affine plane imbedded in it.
Proof: One part of theorem, affine planes being extended to projective planes, is clear from the discussion at the beginning. Now, given a projective plane, pick a line and delete it along with all the points incident to it. It can be easily checked that the system you get is an affine plane.
We denote the projective plane that we get from as . From above theorem, it is clear that we can talk about order of finite projective planes as well.
Theorem 2 Let be a finite projective plane, then there exists a number called the order of the plane such that :
- Each line is incident with points.
- Each point is incident with lines.
- There are lines.
- There are points.
Both affine planes and projective planes are specific examples of a more general structure called an Incidence system.
Def : An incidence system is a triple where is a set of points, a set of blocks and is a binary relation called the incidence relation.
Def : A design is an incidence system with points such that each block is incident with points and any distinct points are incident with exactly blocks.
- An affine plane of order is a design.
- A projective plane of order is a design.
Proof: Let be a point. Let be the number of blocks is incident to. Count the number of ordered pairs where is a point different from and is a block to which both and are incident. First choosing and then we get such pairs. First choosing , then we get such pairs. Therefore, . But the right hand side is a constant, independent of the choice of . Therefore we get a constant given by such that each point is incident to exactly blocks.
Now count the number of ordered pairs where is a block incident to the point . can be chosen in ways and then it has points to which it is incident. Or, we can choose a point first in ways and then there are blocks incident to it. Therefore .
Theorem 4 A design is an affine plane.
Proof: Only the playfair axiom need to be checked as rest all is pretty direct from the definition of a design. Let be a point and a line not incident to it. Since we have , there are lines through . Since doesn’t lie on , each of the points on determine a unique line with . So we are only left with more line passing through . This must be disjoint with and hence parallel to it.
Theorem 5 A design is a Projective plane.
Proof: We just need to check that two lines (blocks) intersect at a unique point. It suffices to show that no two lines are disjoint since if they do intersect then the point must be unique. Let be a line and a point not incident to it. There are points incident to and lines passing through . Each point on determines a unique line along with . Therefore no line through can be parallel (disjoint) to . Hence no two lines can be disjoint.
Therefore, finite affine planes are precisely designs and finite projective planes are precisely designs.
Can we characterize the integer values of for which there is a design? Theorem 3 above does give some necessary conditions that these parameters must satisfy but full characterization is a fundamental and still unresolved problem in design theory. For example, the values , , , , satisfy and but it can be shown that there is no design. Also, we do not know yet if there is a design or not!