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Their attempts were doomed to failure (as we now know, the parallel postulate is not provable from the other postulates), but their efforts led to the discovery of hyperbolic geometry. The theorems of Alhacen, Khayyam and al-Tūsī on quadrilaterals, including the Ibn al-Haytham–Lambert quadrilateral and Khayyam–Saccheri quadrilateral, were ...
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t , sin t ) form a circle with a unit radius , the points (cosh t , sinh t ) form the right half of the unit hyperbola .
In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called sides or edges and three points called angles or vertices . Just as in the Euclidean case, three points of a hyperbolic space of an arbitrary dimension always lie on the same plane.
Intersection theorem (projective geometry) Japanese theorem for concyclic polygons (Euclidean geometry) Japanese theorem for concyclic quadrilaterals (Euclidean geometry) Kawasaki's theorem (mathematics of paper folding) Lester's theorem (Euclidean plane geometry) Lexell's theorem (spherical geometry) Menelaus's theorem ; Miquel's theorem
Hyperbolic 3-manifold; Hyperbolic coordinates; Hyperbolic Dehn surgery; Hyperbolic functions; Hyperbolic group; Hyperbolic law of cosines; Hyperbolic manifold; Hyperbolic metric space; Hyperbolic motion; Hyperbolic space; Hyperbolic tree; Hyperbolic volume; Hyperbolization theorem; Hyperboloid model; Hypercycle (geometry) HyperRogue
Algebraically, hyperbolic and spherical geometry have the same structure. [4] This allows us to apply concepts and theorems to one geometry to the other. [4] Applying hyperbolic geometry to spherical geometry can make it easier to understand because spheres are much more concrete, which then makes spherical geometry easier to conceptualize.
Models of hyperbolic geometry. "By one of the injustices of nomenclature that are so common in mathematics, the three models – which could appropriately be called Riemann-Beltrami, Liouville-Beltrami, and Cayley-Beltrami models – are usually known as the Poincaré disk model , the Poincaré half-plane model and the Klein disk model ."
In hyperbolic geometry, the "law of cosines" is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar law of cosines from plane trigonometry, or the spherical law of cosines in spherical trigonometry. [1] It can also be related to the relativistic velocity addition formula. [2] [3]
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related to: hyperbolic geometry theorems examples list of names and formulas math definition