Definition:Cross-Ratio/Lines through Origin
< Definition:Cross-Ratio(Redirected from Definition:Cross-Ratio of Lines through Origin)
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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
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $20$. Condition that the two pairs of lines through the origin, $y = \lambda x$, $y = \mu x$ and $y = \lambda' x$, $y = \mu' x$ should be apolar
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): cross-ratio
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): cross-ratio