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# Monthly Archives: November 2012

# Lecture 7

# Complete quadrilateral and quadrangle

**Definition (complete quadrilateral).** A configuration consisting of four lines in the projective plane – no three through one point – and the six intersection points, one for each pair of lines, form a *complete quadrilateral*.

# Lecture 6

**Cross-ratio (Doppelverhältnis)**

We are looking for invariants with respect to projective transformations.

**Definition:** Let $ P_{i} =[v_i]= \ssqvector{x_{i} \\ y_{i}}$, $i=1,\ldots,4$, be four distinct points on a projective line $ \RP^{1} $. Then the cross-ratio of these points is

\begin{align*}

cr(P_{1},P_{2},P_{3},P_{4})

&= \frac{\det(v_1v_2)}{\det(v_2v_3)}\frac{\det(v_3v_4)}{\det(v_4v_1)}\\

&= \dfrac{x_{1}y_{2}-x_{2}y_{1}}{x_{2}y_{3}-x_{3}y_{2}} \dfrac{x_{3}y_{4}-x_{4}y_{3}}{x_{4}y_{1}-x_{1}y_{4}}.\\

\end{align*}

If $y_{i} \neq 0$, we may introduce affine coordinates $ u_{i} = \frac{x_{i}}{y_{i}}$. This yields

\begin{align*}

cr(P_{1},P_{2},P_{3},P_{4}) =

&= \dfrac{y_{1}y_{2}( \frac{x_{1}}{y_{1}}-\frac{x_{2}}{y_{2}} ) }{y_{2}y_{3}( \frac{x_{2}}{y_{2}}-\frac{x_{3}}{y_{3}} )} \dfrac{y_{3}y_{4}( \frac{x_{3}}{y_{3}}-\frac{x_{4}}{y_{4}} ) }{y_{4}y_{1}( \frac{x_{4}}{y_{4}}-\frac{x_{1}}{y_{1}} )} \\

&= \dfrac{u_{1}-u_{2}}{u_{2}-u_{3}} \dfrac{u_{3}-u_{4}}{u_{4}-u_{1}}\,.

\end{align*}

# Room changes

Sorry for the inconvenience today. The other dates that we have to leave the regular room are:

**Monday 10.12. MA 649**

**Thursday 13.12. MA 650**

# Notes as single pdf

A pdf version of the notes is maintained by Che Netzer and available for download on his website.

# Lecture 5

# Projective transformations

Let $V$, $W$ be two vectorspaces over the same field and of the same dimension and $F\colon V \rightarrow W$ a linear isomorphism. In particular $ker(F) = \{0\}$, so F maps 1-dimensional subspaces to 1-dimensional subspaces.

Hence $F$ induces a map from $P(V)$ to $P(W)$.

**Definition:** A *projective transformation* $f$ from $P(V)$ to $P(W)$ is a map defined by a linear isomorphism $F\colon V \rightarrow W$ such that

\begin{equation*}

f([v]) = [F(v)] \quad \forall [v] \in P(V)\,.

\end{equation*} Continue reading