Knowra Cross-ratio Cross-ratio The cross-ratio is a scalar associated with four points on a line that remains unchanged under projective transformations. It captures their projective arrangement, including whether the points are harmonic.
Homogeneous coordinates : Coordinates that represent points up to multiplication by a nonzero scalar, enabling projective geometry calculations. They express points on a projective line and make the invariant formula independent of coordinate choice.
Projective line : The set of one-dimensional subspaces of a two-dimensional vector space, geometrically a line with one point at infinity. The cross-ratio is defined for four points on this projective space, including points at infinity.
Von Staudt construction : A synthetic projective construction that defines arithmetic operations using incidence relations among points and lines. It recovers numerical structure from projective constructions, with harmonic division as a key step.
Euclidean distance : The ordinary length of the straight segment joining two points in Euclidean space. Projective maps preserve cross-ratios but generally change distances.
Möbius transformation : A complex function of the form (az+b)/(cz+d), with ad−bc nonzero, acting as a bijection of the extended complex plane. Projective transformations of a line take this fractional-linear form and preserve cross-ratios.
Projective transformation : A map induced by an invertible linear transformation on projective space, preserving lines and incidence. Invariance under these maps is the defining property of the cross-ratio.
Complex analysis : The branch of mathematics studying functions of complex variables and their analytic properties. Cross-ratios encode conformally natural configurations and are preserved by Möbius maps.
Angle : A measure of the rotation or separation between two intersecting rays, commonly expressed in radians or degrees. Angles are not projective invariants, unlike the cross-ratio of four collinear points.
Cross-ratio formula : For four distinct line coordinates, a standard cross-ratio is ((c−a)(d−b))/((c−b)(d−a)). The formula combines pairwise differences so the factors introduced by a projective map cancel.
Projective geometry : The study of geometric properties preserved by projective transformations, especially incidence and alignment. The cross-ratio supplies a numerical invariant where projective geometry otherwise preserves few measurements.
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