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 →

Let $\ell$ be a line in $\RP^2$. Then the line can be described by one homogeneous equation:
$$\begin{bmatrix}
x_1\\
x_2\\
x_3\end{bmatrix}\in l \Longleftrightarrow a_1x_1+a_2x_2+a_3x_3=0\,,$$
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 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 $(\RP^2)^*$.

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