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Cross Ratio

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If a, b, c, and d are points in the extended complex plane C^*, their cross ratio, also called the cross-ratio (Courant and Robbins 1996, p. 172; Durell 1928, p. 73), anharmonic ratio, and anharmonic section (Casey 1888), is defined by

[a,b,c,d]=((a-b)(c-d))/((a-d)(c-b)).
(1)

Here if a=d, the result is infinity, and if one of a, b, c, or d is infinity, then the two terms on the right containing it are cancelled.

For a linear fractional transformation f,

[a,b,c,d]=[f(a),f(b),f(c),f(d)].
(2)

The function f(z)=[z,b,c,d] is the unique linear fractional transformation which takes b to 0, c to 1, and d to infinity. Moreover, f(z) is real if and only if the four points lie on a straight line or a generalized circle.

There are six different values which the cross ratio may take, depending on the order in which the points are chosen. Let lambda=[a,b,c,d]. Possible values of the cross-ratio are then lambda, 1-lambda, 1/lambda, (lambda-1)/lambda, 1/(1-lambda), and lambda/(lambda-1).

Given four collinear points A, B, C, and D, let the distance between points A and B be denoted AB, etc. Then the cross ratio can be defined by

[A,B,C,D]=((AC)(BD))/((AD)(BC)).
(3)

The notation {AB,CD} is sometimes also used (Coxeter and Greitzer 1967, p. 107).

There are a number of different notational and definition conventions for the cross ratio. For example, the definitions [A,B,C,D]=(AB/AD)/(BC/CD) and [A,B,C,D]=(CA/CB)/(DA/DB) are used by Kline (1990) and Courant and Robbins (1966), respectively (Coxeter and Greitzer 1967, p. 107).

The identity

[A,D,B,C]+[A,B,D,C]=1
(4)

holds iff AC//BD, where // denotes separation.

The cross ratio can also be defined for any four coplanar points. It is preserved by any inversion (cf. Ogilvy 1990, p. 40) whose pole is different from any of the given four points (where the last restriction is necessary only to avoid working with infinities).

The following table summarized the cross ratios for some geometric configurations.

figurecross ratio
square ABCD of side length a2
square ABCD of side length a1/2
rectangle ABCD of side lengths a, b1+(a^2)/(b^2)
equilateral triangle DeltaABC with D at center1

See also

Bivalent Range, Equicross, Harmonic Range, Homographic, Möbius Transformation, Separation

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References

Anderson, J. W. "The Cross Ratio." §2.3 in Hyperbolic Geometry. New York: Springer-Verlag, pp. 30-36, 1999.Bogomolny, A. "Cross-Ratio." http://www.cut-the-knot.org/pythagoras/Cross-Ratio.shtml.Casey, J. "Theory of Anharmonic Section." §6.6 in A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 126-140, 1888.Courant, R. and Robbins, H. "Cross-Ratio." What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 172-180, 1996.Coxeter, H. S. M. and Greitzer, S. L. "Cross Ratio." §5.2 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 107-108, 1967.Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 73-76, 1928.Graustein, W. C. "Cross Ratio." Ch. 6 in Introduction to Higher Geometry. New York: Macmillan, pp. 72-83, 1930.Kline, M. Mathematical Thought from Ancient to Modern Times. Oxford, England: Oxford University Press, 1990.Lachlan, R. "Theory of Cross Ratio." Ch. 16 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 266-282, 1893.Möbius, A. F. Ch. 5 in Der barycentrische Calcul: Ein neues Hülfsmittel zur analytischen Behandlung der Geometrie, dargestellt und insbesondere auf die Bildung neuer Classen von Aufgaben und die Entwickelung mehrerer Eigenschaften der Kegelschnitte angewendet. Leipzig, Germany: J. A. Barth, 1827.Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 39-41, 1990.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 41, 1991.

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Cross Ratio

Cite this as:

Weisstein, Eric W. "Cross Ratio." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CrossRatio.html

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