Assignment IV: The Real Projective Plane

Prepared by:

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

email:

malkevitch@york.cuny.edu

web page:

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

1. What are the points of the real projective plane that correspond to the Euclidean points:

a. (-1, 2)

b. (2, -1)

c. (-1, 0)

d. (0, -5)

2. Determine the lines in the real real projective plane which correspond to the Euclidean plane lines:

a. x = 7

b. y = 0

c. y = -9

d. x + y = 11

e. x - y + 4 = 0

3. Determine the points of the Euclidean plane (if they exist) that give rise to the real projective plane points:

a. (0, 3, 5)

b. (-2, 2,, -2)

c. (0, 4, 0)

d. (0, 0, 1)

e. (0, 0, 4)

f, (0, 0, 0)

4. Determine if these Euclidean plane points or real projective plane points are collinear.

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

b. (-1, 3), (2, -7), (2, -8)

c. (-1,2,0), (2,0,0), (2, 3, 0)

d. (0, 0), (2,4), (4,8)

e. (-1, 1, -1), (2, 0, 4), (0,0, 2)

5. Find a line through the point in the Euclidean plane (-2, -1) which is parallel to 2x -3y +1 = 0

6. At what point in the real projective plane do the lines that correspond to the Euclidean lines 2x + 3y - 4 and 2x + 3y - 8 in the real projective plane meet?

7. At what point do the lines x - 9z = 0 and x + 4z = 0 in the real projective plane meet?

8. Draw a "graph" of the points in the Taxicab Plane which:

a. Are taxicab distance equidistant from (0, 0) to (-4, --4).

b. Are taxicab distance equidistant from (0, 0) to (-4, -8).