History of Mathematics
Homework Assignment 4
(Geometry of the Real Projective Plane)
Prepared by:
Joseph Malkevitch
Department of Mathematics and Computer Studies
York College (CUNY)
Jamaica, New York 11451
Given two points in the Euclidean plane, (a, b) and (c, d) we can write down an easy expression for the line which passes through these points using determinants:
Remember that if two rows or columns of a determinant are equal then the value of the determinant is identically 0. Hence, above, if one substitutes for the x and y values x = a and y = b (or x = c and y = d) we get 0. Thus, the above determinant when expanded gives the equation of a line which passes through the points (a, b) and (c, d).
It should not surprise you, therefore, that using homogeneous equations in the real projective plane that:
is the equation of a line that passes through the points (a, b, c) and (e, f, g).
Desargues "statement" is:
Given two triangles a projective plane, ABC and A'B'C',
if:
the lines AA', BB' and CC' pass through a single point
then
given
X = the point where AB and A'B' meet
Y = the point where AC and A'C' meet
Z = the point where BC and B'C' meet
we can conclude that X, Y, and Z lie on a line!
Sometimes this theorem is described by say:
if two triangles are in perspective from a point, they are in perspective from a line.
The converse of this theorem is also true.
If two triangles are in perspective from a line, they are in perspective from a point.
Desargues statement is true in some projective planes and not true in others.
In the real projective plane it is true. It is also true for a projective plane embedded in a 3-dimensional space. Desargues statement is true in some finite projective planes and false in others. An example of an infinite projective plane in which it fails to hold is the Moulton Plane, named for F.R. Moulton. Closely associated with Desargues Theorem is the wonderful theorem of Pappus. Pappus' Theorem (projective plane version) states that given the three points A, B, and C on line l and A', B', and C' on line m that if segments AB' and A'B meet at R, BC' and B'C meet at P, and CA' and C'A meet at Q then P, Q, and R are collinear.
These two theorems were shown to have an unexpected connection with algebra. It turns out that one can introduce a coordinate system for a geometry where the coordinates are drawn from a field provided that Pappus' Theorem holds. One can introduce a system of coordinates where all the requirements of a field except commutativity of multiplication holds provided that Desargues Theorem holds. An algebraic system of this kind is known as a division ring. It turns out that that due to a theorem of Wedderburn, every finite division ring is a field. Thus, in a finite projective plane where Desargues Theorem holds, then Pappus' Theorem holds. Finite non-desarguian planes can not be coordinatized with numbers from a field.
Desargues was French and lived from 1591-1661.
1. Given the point A = (2, 6, 1/2) in the real projective plane:
a. Find other coordinates for this point which are all integers.
b. Find other coordinates for this point which are all negative rational numbers.
c. Find coordinates for A whose first coordinated is 1.
2. If B = (1, -1, 1) and C = (1, 2, 1)
a. Find the equations of the lines AB, AC, and BC.
b. Find the points where each of these lines meets the line at infinity.
3. Given the triangles:
A = (2, 3, 1), B = -1, 3, 1), C = (3, 7, 1)
and
A' = (4, 7, 1), B' = (4, 6, 5), C' = (4, 2, 5)
a. Find the equation of AA'
b. Find the equation of BB'
c. Find the equation of CC'
d. Does these three lines go through a single point? If so give its coordinates.
e. Find the point X where AB and A'B' meet.
f. Find the point Y where AC and A'C' meet.
g. Find the point Z where BC and B'C' meet.
h. Do these three points X, Y, and Z lie on a line?
3. Suppose A = (3, 7, 1), B = (4, 2, 5) and C = (4, 0, 6)
and A' = (-1, 3, 1) and B' = (4, 6, 5) and C' = (1, 7, 4)
a. Determine if A, B, and C are collinear.
b. Determine if A', B', and C' are collinear.
c. If the hypothesis of Pappus' Theorem holds, verify the conclusion.