Lecture 10

Let $\ell$ be a line in $\RP^2$. Then the line can be described by one homogeneous equation:
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 dual projective plane $(\RP^2)^*$. If we fix one point $[x_1,x_2,x_3]^T\in \RP^2$, the set of lines through this point corresponds to a line in $(\RP^2)^*$.

Duality in the real projective plane Continue reading

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

f([v]) = [F(v)] \quad \forall [v] \in P(V)\,.
\end{equation*} Continue reading