Category:Definitions/Cross-Ratios
This category contains definitions related to Cross-Ratios.
Related results can be found in Category:Cross-Ratios.
Points on a Line
Let $A$, $B$, $C$ and $D$ be points on a straight line.
The cross-ratio of $A$, $B$, $C$ and $D$, in that specific order, is defined as:
- $\dfrac {AC / CB} {AD / DB}$
or equivalently:
- $\dfrac {AC \cdot DB} {AD \cdot CB}$
It can be denoted:
- $\set {A, B; C, D}$
Lines through Origin
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:
- $\tuple {\lambda \mu, \lambda', \mu'} := \dfrac {\paren {\lambda - \lambda'} \paren {\mu - \mu'} } {\paren {\lambda - \mu'} \paren {\mu - \lambda'} }$
Complex Analysis
Let $z_1, z_2, z_3, z_4$ be distinct complex numbers.
The cross-ratio of $z_1, z_2, z_3, z_4$ is defined and denoted:
- $\paren {z_1, z_3; z_2, z_4} = \dfrac {\paren {z_1 - z_2} \paren {z_3 - z_4} } {\paren {z_1 - z_4} \paren {z_3 - z_2} }$
Subcategories
This category has only the following subcategory.
H
- Definitions/Harmonic Ratios (2 P)
Pages in category "Definitions/Cross-Ratios"
The following 10 pages are in this category, out of 10 total.
C
- Condition for Cross-Ratio to be Harmonic
- Definition:Cross-Ratio
- Definition:Cross-Ratio of Lines through Origin
- Definition:Cross-Ratio of Lines through Origin/Also presented as
- Definition:Cross-Ratio of Points on Line
- Definition:Cross-Ratio/Also known as
- Definition:Cross-Ratio/Complex Analysis
- Definition:Cross-Ratio/Lines through Origin
- Definition:Cross-Ratio/Points on Line