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isometries in the hyperbolic plane. De Gruyter Studies in mathematics. Vol. 29. Berlin: Walter de Gruyter & Co. Lobachevsky, Nikolai I., (2010) Pangeometry...

Nikolai Ivanovich Lobachevsky (Russian: Никола́й Ива́нович Лобаче́вский, IPA: [nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj] ; 1 December [O.S. 20 November] 1792...

In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted E 2 {\displaystyle {\textbf {E}}^{2}} or E 2 {\displaystyle \mathbb {E}...

In the hyperbolic plane, as in the Euclidean plane, each point can be uniquely identified by two real numbers. Several qualitatively different ways of...

has a variety of properties that differ from those of classical Euclidean plane geometry. For example, the sum of the interior angles of any triangle is...

I. V. Tkachenko 'On trisection and bisection of a triangle in the Lobachevsky plane'] (PDF), Matematicheskoe Prosveschenie, ser. 3 (in Russian), 11: 127–130...

studied, is also called the hyperbolic plane. It is also sometimes referred to as Lobachevsky space or Bolyai–Lobachevsky space after the names of the author...

Tits and geometrically by Ernest Vinberg. In the case of the Lobachevsky or hyperbolic plane, the ideas originate in the nineteenth-century work of Henri...

each pair of planes are mutually parallel. A line can lie in a given plane, intersect that plane in a unique point, or be parallel to the plane. In the last...

of reference. A plane that satisfies Hilbert's Incidence, Betweenness and Congruence axioms is called a Hilbert plane. Hilbert planes are models of absolute...

equations for planes, straight lines, and circles, often in two and sometimes three dimensions. Geometrically, one studies the Euclidean plane (two dimensions)...

were the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation of symmetry...

geometry are finite Möbius or inversive planes and Laguerre planes, which are examples of a general type called Benz planes, and their higher-dimensional analogs...

geometry. Then, in 1829–1830 the Russian mathematician Nikolai Ivanovich Lobachevsky and in 1832 the Hungarian mathematician János Bolyai separately and independently...

tools of spherical trigonometry are in many respects analogous to Euclidean plane geometry and trigonometry, but also have some important differences. The...

A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. The distance between any point of...

study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity...

between two lines (or two line segments), between a line and a plane, and between two planes. Perpendicularity is one particular instance of the more general...

geometry occurred when, around 1830, János Bolyai and Nikolai Ivanovich Lobachevsky separately published work on non-Euclidean geometry, in which the parallel...

{Cl} _{2}(2\theta )/2} though the name "Lobachevsky function" is not quite historically accurate, as Lobachevsky's formulas for hyperbolic volume used the...