cross ratio

cross-ratio, adj.
a ratio relating four points in the real or complex plane. Also called anharmonic ratio.
[1880-85]

* * *

 in projective geometry, ratio that is of fundamental importance in characterizing projections. In a projection of one line onto another from a central point (see Figure—>), the double ratio of lengths on the first line (AC/AD)/(BC/BD) is equal to the corresponding ratio on the other line. Such a ratio is significant because projections distort most metric relationships (i.e., those involving the measured quantities of length and angle), while the study of projective geometry centres on finding those properties that remain invariant. Although the cross ratio was used extensively by early 19th-century projective geometers in formulating theorems, it was felt to be a somewhat unsatisfactory concept because its definition depended upon the Euclidean (Euclidean geometry) concept of length, a concept from which projective geometers wanted to free the subject altogether. In 1847 the German mathematician Karl G.C. von Staudt showed how to effect this separation by defining the cross ratio without reference to length. In 1873 the German mathematician Felix Klein (Klein, Felix) showed how the basic concepts in Euclidean geometry of length and angle magnitude could be defined solely in terms of von Staudt's abstract cross ratio, bringing the two geometries together again, this time with projective geometry occupying the more basic position.

* * *


Universalium. 2010.

Look at other dictionaries:

  • Cross-ratio — D is the harmonic conjugate of C with respect to A and B 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… …   Wikipedia

  • cross ratio — noun Etymology: cross (III) : an anharmonic ratio in mathematics * * * cross ratio, adj. Geom. a ratio relating four points in the real or complex plane. Also called anharmonic ratio. [1880 85] …   Useful english dictionary

  • cross-ratio — cross raˈtio noun Of four points in a range, or rays in a pencil, the quotient of the positions ratios of two with respect to the other two • • • Main Entry: ↑cross …   Useful english dictionary

  • cross-ratio — noun A special number associated with an ordered quadruple of collinear points, particularly points on a projective line, and preserved by the fractional linear transformations. Syn: anharmonic ratio, double ratio …   Wiktionary

  • Cross listing — of one company on multiple exchanges should not be confused with dual listed companies, where two distinct companies with separate stocks listed on different exchanges function as one company. Cross listing of shares is when a firm lists its… …   Wikipedia

  • Cross-sectional study — Cross sectional studies (also known as Cross sectional analysis) form a class of research methods that involve observation of all of a population, or a representative subset, at one specific point in time. They differ from case control studies in …   Wikipedia

  • Cross-validation (statistics) — Cross validation, sometimes called rotation estimation,[1][2][3] is a technique for assessing how the results of a statistical analysis will generalize to an independent data set. It is mainly used in settings where the goal is prediction, and… …   Wikipedia

  • Cross-flow turbine — Cross flow turbine. Image credit; European Communities, Layman s Guidebook (on how to develop a small hydroelectric site) A cross flow turbine, Banki Michell turbine, or Ossberger turbine[1] is a water turbine developed by the Australian Anthony… …   Wikipedia

  • Cross elasticity of demand — Economics …   Wikipedia

  • Cross section (physics) — A cross section is the effective area which governs the probability of some scattering or absorption event. Together with particle density and path length, it can be used to predict the total scattering probability via the Beer Lambert law. In… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.