The Real Projective Plane

prepared by:

Joseph Malkevitch
Department of Mathematics/CS
York College (CUNY)
Jamaica, New York 11451

email:

malkevitch@york.cuny.edu

web page:

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

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. (See the diagram below.)

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. This additional line is usually called the line at infinity. (When we look at parallel lines in the Euclidean plane (like railroad tracks) they appear to meet in the distance, at infinity.)

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 triple (0, 0, 0) does not represent a point.

Here, I have used the variables x, y, and z to represent the coordinates in the real projective plane, but one could use x1, x2, and x3 instead. There are pros and cons of the two different notations.

The lines of our projective plane have the form ax + by +cz = 0, where not all of a, b, and c are 0. (Note that it takes a while to get used to the fact that in this context an equation like 2x + y - z = 0 is a line rather than a plane, as it would be if we were discussing the geometry of 3-dimensional Euclidean space.)

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 the line -2x + y - 4 = 0 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 the line z = 0!

Here is a diagram which you can use see how the real projective plane can be obtained from the Euclidean plane.

The point which is added to all of the lines with the same (or undefined) slope m can be denoted Omega (m) (Omega (undefined)). Thus, Ω-3 (or Ω (-3) can be used to represent a point which lies on all of the lines of slope -3. Remember that (0,0, 0) is not a point in the real projective plane. All the "omega points" satisfy the equation z = 0 in the real projective plane. When a point lies on the line at infinity its last coordinate (z coordinate) is 0.

One form of 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 every 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.

Exercises

Here are some exercises to help you become familiar with the real projective planes.

1. Find the line that passes through the points given:

a. (1, 1, 2) and (2, 3, -1)

b. (0, 2, 1) and (-1, 1, 1)

c. (2, 0, 1) and (-1, 1, 1)

d. (1, 2, 0) and (2, 3, 4)

e. (1, 2, 0) and (2, 3, 0)

2. What Euclidean point, if any, can be associated with the following points in the real projective plane?

a. (2, 3, 1)

b. (-2, -3, 1)

c. (-2, -3, -1)

d. (2, 4, 6)

e. (2, -3, -1)

f. (-1, 2, 0)

g. (1/2, 1/3, 6)

h. (1, 0, 0)

3. What Euclidean line, if any, can be associated with the following lines in the real projective plane?

3. Find the point of the real projective plane where the following lines intersect:

a. x + y + z = 0 and -x + 2y + z = 0

b. 2x - y + 3z = 0 and - x + z = 0

c. -x + z = 0 and y + z = 0

d. x = 0 and y = 0

e. x = 0 and z = 0

f. y = 0 and z = 0

g. -x + 5y - z = 0 and 2x + 7y - 3z = 0.

4. What line of the projective plane can you think of as associated with the following Euclidean lines?

a. x = 7

b. y = 8

c. x + y = 4

d. -x + 3y + 18 = 0

e. 2x - 7y = 21

f. -y + 3x + 5 = 0.

5. Do the following lines all go through a single point? (If so, what point?)

a. x + y + z = 0, 2x - 3y +z = 0, and -x + 7 y - 6 z = 0

b. x + 2y - z = 0, 4x - y = 0, and z = 0.

c. 2x - y + 3z = 0, -4x + 2y + z = 0 and x -(1/2)y + z = 0.

6. Do the following points all lie on a single line? (If so, what line?)

a. (-1, -1, 1), (3, 3, 3) and (1/2, 1/2, 4).

b. (-2, 3, 0) (1, -2, 1) and (2, 0, 0).

c. (1, 1, -1), (2, 0, -2) and (0, 1, 2).