Geometric Structures

Prepared by:

Joseph Malkevitch
Department of Mathematics and Computer Studies
York College (CUNY)
Jamaica, New York 11451

email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk/

Session 3

The real projective plane arises from (plane) Euclidean Geometry as follows:

For each parallel class (e.g. all the lines in this collection do not have a point in common) in Euclidean Geometry create one new point. This point is placed on each of the lines of the original geometry which are in this parallel class. Thus, in the new geometry, each old Euclidean line will have exactly one additional point of the new geometry placed on it. All of the new points created in this way are placed on a single new line. Remember that lines with different slopes are in different parallel classes. (The lines with undefined slope also form a parallel class.) The intuition here is that if one wants to "banish" parallelism one is guided by the appearance of parallel railroad tracks to meet in the distance. Thus, we force lines which are all parallel to meet "far away" on a "line at infinity." The points that are added to each of the parallel classes are thought of as lying on the line at infinity.

In the Euclidean plane the intuition is that as one moves in opposite directions along a line from a point P on the line that one gets to points that are further and further apart. In the real projective plane the intuition is that as one moves in opposite directions along a line from a point P on the line one gets to the same place! Thus, lines in the real projective plane appear to behave like "circles!"

The points of the real projective plane are the old points together with the new points just mentioned. The lines are modified Euclidean lines together with one additional new line.

A nice model for (plane) Euclidean geometry is:

Points are ordered pairs of real numbers (x, y) and lines are equations of the form ax + by +c = 0 where a and b are not both zero. A point P = (x0, y0) is on line l whose equation is ax + by +c = 0 if ax0 + by0 + c = 0.

Note that in the Euclidean plane the points (2, 4) and (1, 2) are different while the lines 2x + 4y = 0 and x + 2y = 0 are the same. (The lines x + 2y + 3 = 0 is also the same as 2x + 4y + 6 =0.) We will see shortly that this "asymmetry" of the Euclidean plane is not the case for the real projective plane.

Here is a nice model for the real projective plane:

Points are ordered triples of real numbers (x, y, z) where the triple (0, 0, 0) is not used. However if k ­ 0 the (kx, ky, kz) represents the same point as (x, y, z).

The lines have the form ax + by +cz = 0, where not all of a, b, and c are 0.

Point P = (x0, y0, z0) is on line ax + by + cz = 0 if ax0 + by0 + cz0 = 0.

Examples:

(-1, 1, 1) is on the line 3x -5y + 8z = 0.

(2,4,6) is the same point as (1, 2, 3) or (1/3, 2/3, 1).

Here is a way to try to help you visualize the real projective plane.

Corresponding to the Euclidean point (x, y) associate the point (x, y, 1) in the real projective plane. Corresponding to the line y = 2x + 4 in the Euclidean, which is the same as the line: -2x + y - 4 = 0, associate the line -2x + y - 4z = 0 in the real projective plane. Note that since this line has slope 2 it will pass through the point (1, 2, 0) in the real projective plane. More generally, all the lines with slope m will pass through the point (1, m, 0). The lines with undefined slope in the Euclidean plane (the so-called vertical lines) have the form: ax + 0y + c = 0 and these lines correspond to the lines ax + 0y + cz = 0 in the real projective plane and they all pass through the point (0, 1, 0) in the real projective plane. The line which goes through points of the form (a, b, 0) in the real projective plane is z = 0!

The axioms of a projective plane are:

1. Given two points there is a unique line which contains the two points.

2. Given two lines there is a unique point which the lines contain. (This means that no lines are ever parallel.)

3. There exist 4 points no three on a line.

One can easily verify that the model above for the real projective plane satisfies these axioms.

The finite projective plane with three points on every line arises from the 6 point finite affine (Euclidean) geometry by exactly the analogous construction to the one just given.

You can also verify that deleting any of the 7 lines from the diagram above results in a plane with 4 points and 6 lines which obeys the Playfair Axiom that there exist unique parallels through any point P not on a line m, parallel to m.