# Exercise sheet 9

In exercise 1, a minus sign was missing. It is to show $\langle x , p \rangle \le -1$ (instead of $\langle x , p \rangle \le 1$).

The exercise sheet has been updated accordingly

# Exercise sheet 8

In the tutorial we discussed doubly ruled quadrics, i.e., quadrics of signature (+,+,-,-)  in $\mathbb{R}P^3$. The term “doubly ruled” expresses the fact that those quadrics contain two families of lines, where each family of lines generates the quadric. Two lines of those families are skew if they are contained in the same family and they intersect if they are contained in different families. Therefore, each point of the quadric can be described as the intersection point of two unique lines, which also span the tangent plane to the quadric at the intersection point.

It is an important fact that any three skew lines in $\mathbb{R}P^3$ determine a unique doubly ruled quadric that contains the three given lines. You may use this statement for the solution of Exercise 8.3.

Happy holidays,

Emanuel

# Exercise Sheet 6

In the exercise sheet 6 exercise 1 you are talking about singular conics. We have never defined these, are these the same as degenerate ones?

# Exercise sheet 5

In exercise 2 on sheet 5, the claim to be proven ($\textrm{cr}(y,p,x,q) = -1$) of course refers to the dual construction. (In the primal construction, these 4 lines are not concurrent).

# Exercise sheet 4

In exercise 3 about the fundamental theorem of affine geometry, it has to be $n > 1$. Otherwise, the statement is obviously false, as for the fundamental theorem of projective geometry.
(The exercise sheet has been updated accordingly)

# Exercise sheet 3

In exercise 3 about projective reflections, one should assume that the point P is not contained in the hyperplane h.
(The exercise sheet has been updated accordingly)