• Thumbnail for 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...
    30 KB (4,844 words) - 22:03, 13 May 2025
  • Thumbnail for Ratio
    triangle. Cross ratio Dilution ratio Displacement–length ratio Dimensionless quantity Financial ratio Fold change Interval (music) Odds ratio Parts-per...
    24 KB (3,714 words) - 08:02, 11 May 2025
  • Thumbnail for Projective harmonic conjugate
    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...
    16 KB (2,611 words) - 10:07, 13 February 2025
  • Thumbnail for Circle
    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....
    46 KB (6,352 words) - 03:48, 7 July 2025
  • 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...
    30 KB (3,641 words) - 21:39, 24 June 2025
  • Cohen's h Cross-ratio Diagnostic odds ratio Forest plot Hazard ratio Likelihood ratio Rate ratio Szumilas M (August 2010). "Explaining Odds Ratios". Journal...
    49 KB (7,070 words) - 18:18, 28 June 2025
  • Thumbnail for Cayley–Klein metric
    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...
    26 KB (3,697 words) - 18:11, 9 July 2025
  • Thumbnail for Projective line over a ring
    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...
    25 KB (2,979 words) - 04:53, 21 March 2025
  • 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
  • Thumbnail for Flag of Switzerland
    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...
    37 KB (4,335 words) - 11:57, 14 June 2025
  • Thumbnail for List of national flags by design
    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...
    90 KB (7,113 words) - 19:58, 8 July 2025
  • Thumbnail for Golden ratio
    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...
    115 KB (13,380 words) - 00:57, 22 June 2025
  • Thumbnail for Truncated triangular trapezohedron
    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...
    9 KB (1,014 words) - 09:57, 17 June 2025
  • Thumbnail for Projective frame
    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...
    5 KB (785 words) - 00:21, 17 March 2025
  • Thumbnail for Projective linear group
    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...
    44 KB (5,613 words) - 10:17, 14 May 2025
  • 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...
    11 KB (1,637 words) - 02:30, 24 June 2025
  • 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...
    6 KB (773 words) - 17:07, 7 December 2024
  • 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...
    90 KB (12,927 words) - 07:18, 30 May 2025
  • 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...
    4 KB (485 words) - 20:00, 2 April 2025
  • Thumbnail for Apollonius quadrilateral
    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...
    4 KB (487 words) - 22:15, 19 April 2025
  • Thumbnail for Golden spiral
    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...
    12 KB (1,373 words) - 18:03, 20 February 2025
  • points are used in the cross ratio which defines the Cayley–Klein metric. Linear fractional transformations leave cross ratio invariant, so any linear...
    12 KB (1,703 words) - 11:09, 1 June 2025
  • Thumbnail for Harmonic quadrilateral
    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...
    4 KB (470 words) - 06:34, 6 April 2025
  • Thumbnail for Pappus's hexagon theorem
    (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...
    17 KB (2,865 words) - 22:17, 19 April 2025
  • 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...
    5 KB (899 words) - 18:10, 4 November 2024
  • projective geometry became understood. The incidence structure and the cross-ratio are fundamental invariants under projective transformations. Projective...
    38 KB (5,099 words) - 22:20, 24 May 2025
  • Thumbnail for Digit ratio
    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...
    64 KB (7,369 words) - 22:09, 22 May 2025
  • 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...
    38 KB (6,298 words) - 19:19, 7 July 2025
  • ". Two types of ratio comparisons can be made, cross-sectional and time-series. Cross-sectional analysis compares the financial ratios of different companies...
    18 KB (1,409 words) - 17:59, 3 April 2025