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,