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Compared to Euclidean geometry, hyperbolic geometry presents many difficulties for a coordinate system: the angle sum of a quadrilateral is always less than 360°; there are no equidistant lines, so a proper rectangle would need to be enclosed by two lines and two hypercycles; parallel-transporting a line segment around a quadrilateral causes ...
Hadwiger's theorem (geometry, measure theory) Helly's theorem (convex sets) Holditch's theorem (plane geometry) John ellipsoid ; Jung's theorem ; Kepler conjecture (discrete geometry) Kirchberger's theorem (discrete geometry) Krein–Milman theorem (mathematical analysis, discrete geometry) Minkowski's theorem (geometry of numbers)
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.
Hyperbolic functions are used to express the angle of parallelism in hyperbolic geometry. They are used to express Lorentz boosts as hyperbolic rotations in special relativity . They also occur in the solutions of many linear differential equations (such as the equation defining a catenary ), cubic equations , and Laplace's equation in ...
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
In the Beltrami-Klein model of the hyperbolic geometry: two ultraparallel lines correspond to two non-intersecting chords. The poles of these two lines are the respective intersections of the tangent lines to the boundary circle at the endpoints of the chords. Lines perpendicular to line l are modeled by chords whose extension passes through ...
One can extend absolute geometry by adding different axioms about parallelism and get incompatible but consistent axiom systems, giving rise to Euclidean and hyperbolic geometry. Thus every theorem of absolute geometry is a theorem of hyperbolic geometry and Euclidean geometry. However the converse is not true.
Cayley's theorem; Clique problem (to do) Compactness theorem (very compact proof) Erdős–Ko–Rado theorem; Euler's formula; Euler's four-square identity; Euler's theorem; Five color theorem; Five lemma; Fundamental theorem of arithmetic; Gauss–Markov theorem (brief pointer to proof) Gödel's incompleteness theorem. Gödel's first ...
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