cross-ratio
Sign in to savethumb|Points , , , and , , , are related by a projective transformation so their cross ratios, and are equal. 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 , , , on a line, their cross ratio is defined as
Article
20 sectionsContents
- Terminology and history
- Definition
- Properties
- Six cross-ratios
- Projective geometry
- Definition in homogeneous coordinates
- Role in non-Euclidean geometry
- Hyperbolic geometry
- Anharmonic group and Klein four-group
- Exceptional orbits
- Transformational approach
- Co-ordinate description
- Ring homography
- Differential-geometric point of view
- Higher-dimensional generalizations
- Volume cross-ratio
- See also
- Notes
- References
- External links
thumb|Points , , , and , , , are related by a projective transformation so their cross ratios, and are equal. 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 , , , on a line, their cross ratio is defined as (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 is the harmonic conjugate of with respect to and precisely if the cross-ratio of the quadruple is , 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.