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...
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triangle. Cross ratio Dilution ratio Displacement–length ratio Dimensionless quantity Financial ratio Fold change Interval (music) Odds ratio Parts-per...
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projective geometry, harmonic conjugacy can also be defined in terms of the cross-ratio as (A, B; C, D) = −1. The four points are sometimes called a harmonic...
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Circle (section Cross-ratios)
indicated circle satisfies the given ratio. A closely related property of circles involves the geometry of the cross-ratio of points in the complex plane....
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Homography (section Cross-ratio)
call the collineation group the automorphism group of the design. The cross-ratio of four collinear points is an invariant under the homography that is...
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Cohen's h Cross-ratio Diagnostic odds ratio Forest plot Hazard ratio Likelihood ratio Rate ratio Szumilas M (August 2010). "Explaining Odds Ratios". Journal...
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Cayley–Klein metric (section Cross ratio and distance)
complement of a fixed quadric in a projective space which is defined using a cross-ratio. The construction originated with Arthur Cayley's essay "On the theory...
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Projective line over a ring (section Cross-ratio)
line points U[0, 1], U[1, 1], U[1, 0] is called the cross-ratio homography. Sometimes the cross-ratio is taken as the value of h on a fourth point x : (x...
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the point at infinity, then the cross-ratio has to be defined by taking the appropriate limit; e.g. the cross-ratio of z 1 , z 2 , z 3 , ∞ {\displaystyle...
70 KB (10,603 words) - 19:06, 8 June 2025
Octahedron (redirect from 3-cross)
3190170508. MR 1242180. Futamura, F.; Frantz, M.; Crannell, A. (2014). "The cross ratio as a shape parameter for Dürer's solid". Journal of Mathematics and the...
13 KB (1,231 words) - 01:36, 20 June 2025
Flag of Switzerland (redirect from Swiss cross)
federal cross. Its arms are equilateral, and their ratio of length to width is 7:6. The size of the cross in relation to the field was set in 2017 as 5:8...
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aspect ratio is 189:335 ≈ 1:1.772 Switzerland The aspect ratio is 1:1 — square-shaped Togo: The aspect ratio is of Fibonacci's golden ratio that is...
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In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed...
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called the cross ratio. Their estimate of the cross ratio is close to MacGillavry's, and has a numerical value close to the golden ratio. Based on this...
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coordinates of a point a ≠ a0 are (r, 1), where r is the cross-ratio (a, a2; a1, a0). If a = a0, the cross ratio is the infinity, and the projective coordinates...
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and is the unique such map that does so. This is the cross-ratio (x, b; a, c) – see Cross-ratio § Transformational approach for details. For n ≥ 3, PGL(n...
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calculation can be achieved by considering the ratios as those of similar triangles. In practice, the method of cross-multiplying means that we multiply the numerator...
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monohybrid cross to create the dihybrid cross. From these experiments, he determined the phenotypic ratio (9:3:3:1) seen in dihybrid cross for a heterozygous...
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the cross ratio invariant: (g(a),g(b);g(c),g(d)) = (a,b:c,d). Since the Möbius group acts simply transitively on triples of points, the cross ratio can...
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making cross-sector comparisons less reliable. Financial ratio Price-to-earnings ratio Price-to-cash-flow ratio "Price-to-Sales Ratio (P/S Ratio)". Investopedia...
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definition is that the cross ratio of the four points is ± 1 {\displaystyle \pm 1} . It is allowed for the quadrilateral sides to cross. The Apollonius quadrilaterals...
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Golden spiral (category Golden ratio)
the projective harmonic conjugate of B with respect to A, D, i.e. the cross ratio (A,D;B,C) has the singular value −1. The golden spiral is the only logarithmic...
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points are used in the cross ratio which defines the Cayley–Klein metric. Linear fractional transformations leave cross ratio invariant, so any linear...
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square and the cross ratio of its vertices. Four points in the complex plane define a harmonic quadrilateral when their complex cross ratio is − 1 {\displaystyle...
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(KJ/AG)(AG/JL) = (JD/GD)(BG/JB). The last compound ratio (namely JD : GD & BG : JB) is what is known today as the cross ratio of the collinear points J, G, D, and B...
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circular points, natural logarithm and cross-ratio: The angle between two lines is the logarithm of the cross-ratio of the pencil formed by the two lines...
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projective geometry became understood. The incidence structure and the cross-ratio are fundamental invariants under projective transformations. Projective...
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The digit ratio is the ratio taken of the lengths of different digits or fingers on a hand. The most commonly studied digit ratio is that of the 2nd (index...
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Pentagram map (section Cross-ratio)
authors consider 1/X to be the cross-ratio, and that is why X is called the inverse cross ratio. The inverse cross ratio is invariant under projective...
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". Two types of ratio comparisons can be made, cross-sectional and time-series. Cross-sectional analysis compares the financial ratios of different companies...
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