End of term get-together

Hello everybody,

Tomorrow, Thursday the 14th, we will meet at 7 pm at “Cafe A” to have a good time together and welcome the end of the semester. Cafe A is the students cafe at the ground floor of the architecture building. http://goo.gl/maps/i7aWJ

Everybody is welcome and I’m looking forward to seeing you tomorrow,
Emanuel

Tutorials 9./10. 1.

Hello everybody,

I just realized that I was talking nonsense a week ago, claiming the projective metric would be given by
$$\cosh d_{pr} (P,Q) = \frac{1}{2} \ln \cr(Q,X,P,Y)$$.
The correct formula is
$$d_{pr} (P,Q) = \frac{1}{2} \ln \cr(Q,X,P,Y)$$.
I hope you realized that for yourself in the meantime ūüėČ

Have a nice week-end,
Emanuel

Tangent planes to quadrics

In the tutorials, we revisited that for a curve that satisfies $\langle \gamma , \gamma \rangle = c$ one always has $\langle \gamma’ , \gamma \rangle = 0$, i.e., $\gamma \perp \gamma’$ with respect to the product $\langle \cdot , \cdot \rangle$.

This means that tangent vectors $v$ to the quadric $Q = \left\{ x \mid \langle x , x \rangle = c \right\}$ at a point $p \in Q$ are characterized by $\langle p , v \rangle = 0 \Leftrightarrow v \in p^\perp$. However, the affine plane that is tangent to $Q$ at $p$ is given by $\left\{ x = p + v \mid v \in p^\perp \right\} = \left\{ x \mid \langle x , p \rangle = \langle p , p \rangle = c \right\}$.

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