Definitions for cross-ratio
cross-ratio

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1. cross-rationoun

   A special number associated with an ordered quadruple of collinear points, particularly points on a projective line, and preserved by the fractional linear transformations.

WikipediaRate this definition:0.0 / 0 votes

1. Cross-ratio

   In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points A, B, C and D on a line, their cross ratio is defined as ( A , B ; C , D ) = A C ⋅ B D B C ⋅ A D {\displaystyle (A,B;C,D)={\frac {AC\cdot BD}{BC\cdot AD}}} where an orientation of the line determines the sign of each distance and the distance is measured as projected into Euclidean space. (If one of the four points is the line's point at infinity, then the two distances involving that point are dropped from the formula.) The point D is the harmonic conjugate of C with respect to A and B precisely if the cross-ratio of the quadruple is −1, called the harmonic ratio. The cross-ratio can therefore be regarded as measuring the quadruple's deviation from this ratio; hence the name anharmonic ratio. The cross-ratio is preserved by linear fractional transformations. It is essentially the only projective invariant of a quadruple of collinear points; this underlies its importance for projective geometry. The cross-ratio had been defined in deep antiquity, possibly already by Euclid, and was considered by Pappus, who noted its key invariance property. It was extensively studied in the 19th century.Variants of this concept exist for a quadruple of concurrent lines on the projective plane and a quadruple of points on the Riemann sphere. In the Cayley–Klein model of hyperbolic geometry, the distance between points is expressed in terms of a certain cross-ratio.

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1. Cross-ratio

   In geometry, the cross-ratio, also called double ratio and anharmonic ratio, is a special number associated with an ordered quadruple of collinear points, particularly points on a projective line. Variants of this concept exist for a quadruple of concurrent lines on the projective plane and a quadruple of points on the Riemann sphere. The cross-ratio is preserved by the fractional linear transformations and it is essentially the only projective invariant of a quadruple of points, which underlies its importance for projective geometry. In the Cayley–Klein model of hyperbolic geometry, the distance between points is expressed in terms of a certain cross-ratio. Cross-ratio had been defined in deep antiquity, possibly already by Euclid, and was considered by Pappus, who noted its key invariance property. It was extensively studied in the 19th century.

How to pronounce cross-ratio?

1. Alex

   US English

   David

   US English

   Mark

   US English

   Daniel

   British

   Libby

   British

   Mia

   British

   Karen

   Australian

   Hayley

   Australian

   Natasha

   Australian

   Veena

   Indian

   Priya

   Indian

   Neerja

   Indian

   Zira

   US English

   Oliver

   British

   Wendy

   British

   Fred

   US English

   Tessa

   South African

How to say cross-ratio in sign language?

Numerology

1. Chaldean Numerology

   The numerical value of cross-ratio in Chaldean Numerology is: 6

2. Pythagorean Numerology

   The numerical value of cross-ratio in Pythagorean Numerology is: 2

References

1. ^ Wiktionary
   https://en.wiktionary.org/wiki/Cross-Ratio
2. ^ Wikipedia
   https://en.wikipedia.org/wiki/Cross-Ratio
3. ^ Wikidata
   https://www.wikidata.org/w/index.php?search=cross-ratio