Cross-ratio of lines through a point (again).

**Claim.** The map which maps a point $g^{\ast} \in p^{\ast}$ to the intersection point of $g \cap \ell$ is a projective map. Continue reading

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Cross-ratio of lines through a point (again).

**Claim.** The map which maps a point $g^{\ast} \in p^{\ast}$ to the intersection point of $g \cap \ell$ is a projective map. Continue reading

$$\begin{bmatrix}

x_1\\

x_2\\

x_3\end{bmatrix}\in l \Longleftrightarrow a_1x_1+a_2x_2+a_3x_3=0\,,$$

where the $a_i$ are unique up to a scalar multiple $\lambda\neq0$. We can take in a way that will be explained in detail, the point $[a_1,a_2,a_3]^T$ as homogeneous coordinates for the line $\ell$. The lines in $\RP^2$ yield another projective plane with homogeneous coordinates $[a_1,a_2,a_3]^T$. This is what we call the

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**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*.

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**

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

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

I put the configurations I showed on the projector online:

You may move the points around to see how the configuration changes.