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The cross ratio of the points  ,  ,  , and  in
 is denoted by
![$ [a, b, c, d\,]$](http://images.planetmath.org:8080/cache/objects/7223/l2h/img6.png "$ [a, b, c, d\,]$") and is defined by
Some authors denote the cross ratio by
 .
Example 1 The cross ratio of  ,  ,  , and  is
Example 2 The cross ratio of  ,  ,  , and  is
- The cross ratio is invariant under Möbius transformations and projective transformations. This fact can be used to determine distances between objects in a photograph when the distance between certain reference points is known.
- The cross ratio
![$ [a, b, c, d\,]$](http://images.planetmath.org:8080/cache/objects/7223/l2h/img19.png "$ [a, b, c, d\,]$") is real if and only if  ,  ,  , and  lie on a single circle on the Riemann
sphere.
- The function
 defined by
is the unique Möbius transformation which sends  to  ,  to 0, and  to  .
- 1
- Ahlfors, L., Complex Analysis. McGraw-Hill, 1966.
- 2
- Beardon, A. F., The Geometry of Discrete Groups. Springer-Verlag, 1983.
- 3
- Henle, M., Modern Geometries: Non-Euclidean, Projective, and Discrete. Prentice-Hall, 1997 [2001].
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"cross ratio" is owned by rspuzio. [ full author list (2) | owner history (1) ]
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(view preamble)
Cross-references: function, Riemann sphere, circle, lie on, real, objects, distances, projective transformations, Möbius transformations, invariant, points
There are 3 references to this entry.
This is version 5 of cross ratio, born on 2005-07-14, modified 2006-12-08.
Object id is 7223, canonical name is CrossRatio.
Accessed 2359 times total.
Classification:
| AMS MSC: | 30C20 (Functions of a complex variable :: Geometric function theory :: Conformal mappings of special domains) | | | 51N25 (Geometry :: Analytic and descriptive geometry :: Analytic geometry with other transformation groups) | | | 30F40 (Functions of a complex variable :: Riemann surfaces :: Kleinian groups) |
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Pending Errata and Addenda
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![$\displaystyle [a, b, c, d\,] = \frac{a-c}{a-d}\cdot\frac{b-d}{b-c}. $](http://images.planetmath.org:8080/cache/objects/7223/l2h/img7.png "$\displaystyle [a, b, c, d\,] = \frac{a-c}{a-d}\cdot\frac{b-d}{b-c}. $")
![$\displaystyle T(z) = [z, b, c, d\,] $](http://images.planetmath.org:8080/cache/objects/7223/l2h/img25.png "$\displaystyle T(z) = [z, b, c, d\,] $"){kind=small}