In mathematics, a Cayley–Klein metric is a metric on the complement of a fixed quadric in a projective space which is defined using a cross-ratio. The...

In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of hyperbolic geometry...

three dimensions SO(4) 3D rotation group Beltrami–Klein model (Cayley–Klein model) Cayley–Klein metric Goldstein (1980) considers a passive (contravariant...

algebra Cayley–Klein metric Cayley–Klein model of hyperbolic geometry Cayley–Menger determinant Cayley–Purser algorithm Cayley's formula Cayley's hyperdeterminant...

and non-Euclidean geometries could be considered metric spaces determined by a Cayley–Klein metric. This insight had the corollary that non-Euclidean...

form. The Cayley–Klein metrics provided working models of hyperbolic and elliptic metric geometries, as well as Euclidean geometry. Klein is responsible...

geometry, and Felix Klein, in several publications, established the field of non-euclidean geometry through the use of the Cayley-Klein metric. The idea of an...

Cayley's formula Cayley–Klein metric Cayley–Klein model of hyperbolic geometry Cayley's Ω process Cayley surface Cayley transform Cayley's nodal cubic surface...

points. These subsets of the complex plane are provided a metric with the Cayley–Klein metric. Then the distance between two points is computed using the...

circle is the Cayley absolute that determines a metric on the disk through use of cross-ratio in the style of the Cayley–Klein metric. In the language...

manifold. One application of the real projective line is found in Cayley–Klein metric geometry. R4 can be imagined using the fact that 16 points (x1, x2...

the unit disc to itself. The distance between points is given by a Cayley–Klein metric, known to be invariant under the translations since it depends on...

Klein's icosahedral cubic surface Klein's j-invariant Beltrami–Klein model Cayley–Klein metric Clifford–Klein form Schottky–Klein prime form Klein's Encyclopedia...

are recounted. Elliptic geometry according to Clifford, and the Cayley-Klein metric are mentioned to illustrate non-Euclidean geometry. There is an anticipation...

disk, plus all diameters of the disk. Distances in this model are Cayley–Klein metrics. Given two distinct points p and q inside the disk, the unique hyperbolic...

of Euclidean geometry, with the Cayley–Klein metric. The use of a symmetric matrix for a von Staudt conic and metric, applied to screws, has been described...

the advent of special relativity it was used in topics such as the Cayley–Klein metric, hyperboloid model and other models of hyperbolic geometry, computations...

theory and projective geometry, formulating concepts such as the Cayley–Klein metric or the hyperboloid model in which the interval x 1 2 + x 2 2 + x...

divisibility of the whole.": 181  To explain Meinong, Russell refers to the Cayley–Klein metric, which uses stretch coordinates in anharmonic ratios which determine...

geometry in the complex projective line. These models are instances of Cayley–Klein metrics. The cross-ratio may be defined by any of these four expressions:...

with their standard metrics, along with hyperbolic space. The complex projective space, quaternionic projective space, and Cayley plane are analogues...

1859, Arthur Cayley initiated a unification of metric geometries through use of the Cayley-Klein metrics. Later Felix Klein used such metrics to provide...

Bacaloglu develops the "Bacaloglu pseudosphere". Arthur Cayley produces the first Cayley–Klein metric. Bernhard Riemann publishes his paper on number theory...

them is the measure of the angle POQ, usually taken in radians. Arthur Cayley initiated the study of elliptic geometry when he wrote "On the definition...

define a metric. The projective transformations that leave the conic section or quadric stable are the isometries. "Klein showed that if the Cayley absolute...

by David Hilbert (1895) as a generalization of Cayley's formula for the distance in the Cayley–Klein model of hyperbolic geometry, where the convex set...

are the same in the model as they are in the actual hyperbolic plane. The Cayley transform provides an isometry between the half-plane model and the Poincaré...

them is the measure of the angle POQ, usually taken in radians. Arthur Cayley initiated the study of elliptic geometry when he wrote "On the definition...

the Fellowship Examination of Trinity College) which discussed the Cayley–Klein metrics used for non-Euclidean geometry. He attended the first International...

Millennium Prize Problem. Geometric group theory often revolves around the Cayley graph, which is a geometric representation of a group. Other important topics...