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Topic: Cross-ratio

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PlanetMath: cross ratio This is version 4 of cross ratio, born on 2005-07-14, modified 2005-08-02. 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. www.planetmath.org /encyclopedia/CrossRatio.html

PlanetMath: odds ratio This is version 5 of odds ratio, born on 2004-10-04, modified 2005-01-12. It turns out that the log-odds ratio has an asymptotic normal distribution, which can be used to find the confidence interval of OR. On the other hand, the closer to 1 the odds ratio is, the closer to independence between the variables. planetmath.org /encyclopedia/OddsRatio.html

The Hindu Business Line : Banks must cross-sell for retail asset explosion Cross selling is a team effort and success depends on the attitude and involvement of all the staff concerned. Cross selling is a team effort and success depends on the attitude and involvement of all concerned. In the present day retail banking scenario, systematic cross selling of assets is happening in some banks, but in most others, an integrated approach to retail asset expansion through cross selling is still at a nascent stage. www.thehindubusinessline.com /2004/09/14/stories/2004091400100900.htm

Projective Geometry The cross ratio of a given pencil of points is equal to the cross ratio of any pencil of lines related to the pencil of points by a perspectivity. The cross ratio of a given pencil of lines is equal to the cross ratio of any pencil of points related to the pencil of lines by a perspectivity. The cross ratio may be used as an indicator of how an object divides space in a perspective drawing. www.bsu.edu /web/mdlade/Project

Managing Sunflower Price Risk For canola futures, a cross-hedge ratio of.85 was estimated (91 percent effective), and for soybean oil futures, a cross-hedge ratio of.16 was estimated (84 percent effective). For canola futures, a cross-hedge ratio of.85 was estimated, and for soybean oil futures, a cross-hedge ratio of.16 was estimated. In this cross-hedge, a cross hedge ratio of.97 was calculated for canola (92 percent effective) and.30 for soybean oil (88 percent effective). www.ext.nodak.edu /extpubs/agecon/market/ec1270w.htm

RMK articles : Two kinds of Cross Ratio Probabilities -- Preliminary When one cross ratio is evaluated with value k, the other cross ratios are known immediately, and have values (1-k), (1-1/k), 1/k,1/(1-k), and 1/(1-1/k). The concept is further strengthened by the recognition that Fermi's golden rule of transition probability is one of these cross ratios that is bounded by zero and 1. One of the things about projective geometry that has fascinated me for more than thirty years is the fact that all (projective) invariants are cross ratios (or functions thereof), and that except for special cases there exist six fundamental cross ratios for any configuration. www22.pair.com /csdc/pd2/pd2fre47.htm

cross ratio --  Encyclopædia Britannica The piston displacement is the volume swept by the piston during one stroke and is equal to the cross-sectional area of the cylinder multiplied by the length of the stroke. The cross recalls the Crucifixion of Jesus and the redeeming benefits of his Passion and death. 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. www.britannica.com /eb/article?tocId=9028000

Cross-ratio Historically it was noticed that if lines in the plane pass through a P and a fifth line L not P crosses them in four points then cross-ratio of the directed lengths on L by the four points taken in order independent of L. That is it is invariant of the system of four lines. This fact is a special case of fact that cross-ratios are preserved by projective For that reason they play an important in projective geometry. The cross-ratio is real if and only the four complex numbers lie either on common line or on a common circle. www.freeglossary.com /Cross-ratio

Definitions and Measures of the Cross Ratio Draw a line, choose points A and B. Choose points C and D on AB so that they divide AB in nice ratios and so give nice numbers for the parameters m and n etc. Evaluate the vector form of the cross ratio and compare this with the evaluation of the signed segment definition. Use the complex number definition to evaluate the cross ratio for a pencil of 4 points relative to some coordinate axis. Choose 4 points which are concyclic and use the complex number definition to evaluate the cross ratio. s13a.math.aca.mmu.ac.uk /Geometry/M23Geom/XRatio/xratio.html

Cross-ratio - Wikipedia, the free encyclopedia Historically it was noticed that if four lines in the plane pass through a point P, and a fifth line L not through P crosses them in four points, then the cross-ratio of the directed lengths on L formed by the four points taken in order was independent of L. Java Applet demonstrating the invariance of the cross ratio unter a bilinear transformation Cross-ratios are invariants of projective geometry in the sense that they are preserved by projective transformations. en.wikipedia.org /wiki/Cross-ratio

Forest Science: Influence of vertical foliage structure on the distribution of stem cross-sectional area increment in western hemlock and balsam fir [For. Sci. 46(1):86-94] / Reply The original pipe model of stem form is based on observations of an approximately constant ratio between the crosssectional area of a stem or branch and the total mass of foliage borne by that stem or branch (Shinozaki et al. However, a constant ratio of cross-sectional area to leaf mass does not imply a constant ratio of cross-sectional area to leaf area within the crown of a tree. That the ratio proved to be decidedly variable, declining from the top to the bottom of a crown, could be viewed as a serious disagreement between theory and reality. www.findarticles.com /p/articles/mi_qa4019/is_200102/ai_n8952557

Plane-5.htm To calculate the cross ratio, project these points onto the x-axis, which is the line y = 0 or [0,1,0], from the point at infinity on the y-axis, which is (0,1,0). The cross ratio of these four numbers is (3-4)(1-0)/(3-0)(1-4) = (4)(1)/(3)(2) = 4/1 = 4. Projecting onto the y-axis gives the numbers 0,2,3,4 whose cross ratio is (3-0)(4-2)/(3-2)(4-0) = (3)(2)/(1)(4) = 1/4 = 4. www.math.fau.edu /Richman/geometry/Plane-5.htm

Definition of cross 10:...t through P crosses them in four points, then the cross-ratio of the directed lengths on L formed by the... However, cross posting to groups that are irrelevant to the mess... Not to be confused with ''[[Blue Cross]]'', which is a health care plan for humans in th... www.wordiq.com /search/cross.html

World Flag Database: Red Cross It adopted the red cross on a white field as the emblem of societies working for the relief of wounded soldiers. A cross, crescent, star of David, etc can be placed within the chevrons for some uses. An initial proposal for a red diamond was not considered suitable and the latest propsal is for two chevrons, one inverted. www.flags.net /old/ICRC.htm

VIVA (Viewpoint Invariant Visual Acquisition) The cross ratio is chosen because it is the simplest and most fundamental projective invariant. The measured cross ratio can be used as an index into the database because the cross ratio is unchanged under projection to an image. The effectiveness of the cross ratio in model based vision is being assessed within VIVA. www.esat.kuleuven.ac.be /~konijn/news2.2.html

Dihybrid Cross Problem Set A genetic cross yielding a 9:3:3:1 ratio of offspring. Our objective is to understand the principles that govern inheritance of different traits in a dihybrid cross that led Mendel to propose that alleles of different genes are assorted independently of one another during the formation of gametes. Mendel invented the dihybrid cross to determine if different traits of pea plants, such as flower color and seed shape, were inherited independently. www.biology.arizona.edu /mendelian_genetics/problem_sets/dihybrid_cross/dihybrid_cross.html

Cross ratios and harmonic conjugate elements From previous conclusion we can state: The cross ratio (a,b,c,d) is by definition the cross ratio of the intersection points of the ordered quartet of lines with an arbitrary line. Extending the notion of cross ratio to projective level. But a cross ratio = 1 is impossible for a quartet of points. www.ping.be /%7Eping1339/cross.htm

POINback-1.html Cross ratios have several symmetry properties which ensure that if the cross ratio of four points is known, so is the cross ratio of any permutation of these points. and define the cross ratio of complex numbers: the cross ratio of any four distinct finite points I begin with a review of complex numbers and their properties (essential since all the calculations and proofs of the course are based on complex numbers, and the students are usually weak in this area). www.cecm.sfu.ca /~jalester/DEMO/POINCARE/DemoInfo/POINback-1.html

Photograph Puzzles Now set up a cross ratio; in the special case D is a point at infinity, just take a limit and get the cross ratio of A,B,C,D to equal (CA/CB). The cross ratio is such a quantity; given points A,B,C,D on a line, their cross ratio is (CA/CB)/(DA/DB). If A2,B2,C2,D2 are the projections of A,B,C,D respectively, then the cross ratio of these points is equal to the cross ratio of A,B,C,D. For a proof, see Figure 2. www.geom.uiuc.edu /docs/forum/photo

Cross-linked Allophycocyanin ratio obtained in the presence of 1 M sodium perchlorate) is a measure of the effectiveness of cross-linking. Record the ratio after 30 minutes at room temperature. www.prozyme.com /technical/apcxldata.html

Demonstration of Cross Ratio in showing that the cross ratio is unaffected by projection. What we want to show is that the cross ratio (In this demonstration, we will only consider finite points, although the cross ratio holds for infinite points as well.) Define the distance ai.stanford.edu /~birch/projective/node25.html

Citations: Projective area-invariants as an extension of the cross-ratio - Nielsen, Sparr (ResearchIndex) To conclude, the results of this section provide parametrizations of the projective plane and space, similar to the one obtained from the cross ratio on the projective plane. ....are noisy measurements of the projection of four points with a given cross ratio. Other such invariants, generalizing the cross ratio but based on areas, have been developed earlier by the authors, c. citeseer.ist.psu.edu /context/94337/0

SSM Online - News Researchers measured the ratio of cross-matched blood (C) to transfused blood (T) - the C/T ratio. The standard C/T ratio for this surgery is 2. "A high C/T ratio means more blood must be kept in inventory, which increases hospital costs, wastes personnel time, and increases the likelihood of outdated blood products," says Kassell. www.ssmonline.org /News/ViewRelease.asp?ReleaseID=1971

Estimació de l'error comès en determinar ... Then we calculate the cross ratios of two sets of four verticals of the buildings in the photograph. The geometric locus of the points X and X' with these cross ratios on the map are two conics whose intersection is the searched point. According to the Chasles theorem, the geometric locus of the points X for which the cross ratio of the lines XA, XB, XC and XD is constant (for four given points A, B, C and D): www.terra.es /personal/rgonzal1/photo.htm

c To solve a practical problem involving ratios, we use cross-multiplying (an application of the properties of proportional ratios). A ratio is a comparison of two or more numbers. A percentage is a ratio: 10% is 10 parts per hundred or the ratio 10/100 www.uwec.edu /faulkndj/G104/g104fundls/c.htm

Calculation of the cross ratio It is known fact in function theory that the cross ratio is invariant in linear (Möbius) transformations, that is, functions of form After some algebraic manipulations, the cross ratio calculated for particular generators can be expressed in the form (we assume that all generators of the set are know numerically) The double ratios and resulting polynomials were calculated in Mathematica, and the calculation of the roots and plotting were done with the contour-plot routine. www.csc.fi /math_topics/DH/node7.html

Cross Ratio Observe the three sets of points for which the cross ratio is computed and their relationship to one another. The cross ratio may be use as an indicator of how an object divides space. In this appellate, the cross ratio is used to examine Alberti's Method. www.bsu.edu /web/mdlade/Project/CROSSRAT.HTM

Complex_Cross_Ratio A special case of quadrangles, the so called [harmonic], which are related to the cross ratio, are studied in the file HarmonicQuad.html. Below the two cross ratios are calculated using the User-Tool [ComplexCrossRatio] found and compiled in the file [EUC_Scripts\EUC_User_Tools\ComplexCrossRatio.txt]. Concerning the proof, notice that the ratio AC/BC is equal to the ratio of sines: sin(AXC)/sin(CXB), and this is independent of the position of point X on the circumference. www.math.uoc.gr /~pamfilos/eGallery/problems/Complex_Cross_Ratio.html

Basics The cross ratio of four points is the only numerical invariant of projective geometry (if it can be related to Euclidean space). Flat line pencils and axial pencils of planes containing a common line also have cross ratios. The ratios of the distances of successive points from the other invariant point are constant. www.anth.org.uk /NCT/basics.htm

AllRefer.com - projective geometry (Mathematics) - Encyclopedia Two properties that are invariant under projection are the order of three or more points on a line and the harmonic relationship, or cross ratio, among four points, A, B, C, D, i.e., AC/BC : AD/BD. However, lengths and ratios of lengths are not invariant under projection, nor are angles or the shapes of figures. The concept of parallelism does not appear at all in projective geometry; any pair of distinct lines intersects in a point, and if these lines are parallel in the sense of Euclidean geometry, then their point of intersection is at infinity. reference.allrefer.com /encyclopedia/P/projctgeo.html

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