Definition:Cross-Ratio/Lines through Origin

From ProofWiki
Jump to navigation Jump to search

Definition

Let $\LL_1$, $\LL_2$, $\LL_3$ and $\LL_4$ be lines through the origin $O$ whose equations embedded in the Cartesian plane are as follows:

\(\ds \LL_1: \ \ \) \(\ds y\) \(=\) \(\ds \lambda x\)
\(\ds \LL_2: \ \ \) \(\ds y\) \(=\) \(\ds \mu x\)
\(\ds \LL_3: \ \ \) \(\ds y\) \(=\) \(\ds \lambda' x\)
\(\ds \LL_4: \ \ \) \(\ds y\) \(=\) \(\ds \mu' x\)


The cross-ratio of $\LL_1$, $\LL_2$, $\LL_3$ and $\LL_4$, in that specific order, is defined and denoted:

$\tuple {\lambda \mu, \lambda', \mu'} := \dfrac {\paren {\lambda - \lambda'} \paren {\mu - \mu'} } {\paren {\lambda - \mu'} \paren {\mu - \lambda'} }$


Also presented as

Some sources present the cross-ratio of lines through the origin as:

$\tuple {\lambda \mu, \lambda', \mu'} := \dfrac {\lambda - \lambda'} {\lambda - \mu'} / \dfrac {\mu - \lambda'} {\mu - \mu'}$


Also known as

Some sources do not hyphenate cross-ratio, leaving it as: cross ratio.


Also see

  • Results about cross-ratios can be found here.


Sources