Interactive Course on Projective Geometry


Exercises -- First set

1.[00] If a projective transformation of the real projective line sends the point 1 onto the point 2, the point 2 onto the point 3, and the point 3 onto the point 1, where does it send the point 0?

2.[15] A perspective transformation is a projective transformation from RP1 to itself that has a fixed point. Show that any projective transformation is a composition of at most three perspective transformations.

3.[00] A complex projective transformation fixes the unit circle and sends 1/2 onto i/4. Where does it send 2?

In the following two problems C denotes a smooth, strictly convex curve in R2 that is centrally symmetric and centered at the origin. The transformation TC from the projective line to itself is defined by sending the line L to the unique line which passes through the origin and is parallel to the lines tangent to C at the points of intersection of C and L.

4.[10] Show that if C is an ellipse, the transformation TC is projective.

5.[25] Show that if TC is projective, then C is an ellipse.

6.[10] Let a, b, and c be three distinct straight lines through the origin in R2 and let v and w be two nonzero vectors on a and b. Show that if R is the linear transformation which fixes v and takes w to -w, then [a,b,c,R(c)] = -1.


Exercises -- Second set

1.[05] Let a,b,m,n, and p be five distinct points in the real projective line. Show that [a,b,m,n] [a,b,n,p] [a,b,p,m] = 1.

2.[05] Show that

  1. [a,b,c,d] = [b,a,c,d]-1 = [a,b,d,c]-1.
  2. [a,b,c,d] + [a,c,b,d] = 1.

3.[10] Use the previous exercise to show that if [a,b,c,d] =: k, then the only values that it is possible to obtain as the cross ratio of a permutation of a,b,c,d are: k, 1/k, 1-k, 1 - 1/k, 1/(1-k), and k/(k-1).

4. [15] In the following figure, the lines ED and FD are tangent to the conic. Show that [A,B,C,D] = -1 .

5.[20] Let H be the point on the line AB such that [A,E,B,H] = -1, let I be the point on the line BC such that [F,C,B,I] = -1, and let J be the point on the line AC such that [A,C,G,J] = -1. Show that H, I, and J are collinear.