Lecture 9

Fundamental Theorem of real projective geometry: Let $f\colon \RP^n \to \RP^n$, $n \ge 2$, be a bijective map that maps lines to lines. Then $f$ is a projective transformation.

Remark: The theorem does not hold for arbitrary fields. For example $f\colon \CP^n \to \CP^n$ ($n \ge 2$) with
\[f\left(\sqvector{z_1\\ \vdots\\ z_{n+1}}\right) = \sqvector{\bar z_1\\ \vdots\\ \bar z_{n+1}}\]
is a bijective map, mapping complex projective lines to complex projective lines but it is not a projective transformation of $\CP^n$.

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Lecture 4

Definition: Two triangles $\triangle_1 = \triangle(A_1, B_1, C_1)$ and $\triangle_2 = \triangle(A_2, B_2, C_2)$ are in perspective w.r.t. a point $S$ if

S = (A_1 A_2) \cap (B_1 B_2) \cap (C_1 C_2)

The triangles are in perspective w.r.t. a line $\ell$ if

A’ = (B_1 C_1) \cap (B_2 C_2)\\
B’ = (A_1 C_1) \cap (A_2 C_2)\\
C’ = (A_1 B_1) \cap (A_2 B_2)\\

lie on $\ell$.

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