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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 ...
Download as PDF; Printable version; In other projects ... move to sidebar hide. Hyperbolic theory may refer to: Hyperbolic geometry; The theory of hyperbolic partial ...
Hyperbolic geometry is a non-Euclidean geometry where the first four axioms of Euclidean geometry are kept but the fifth axiom, the parallel postulate, is changed.The fifth axiom of hyperbolic geometry says that given a line L and a point P not on that line, there are at least two lines passing through P that are parallel to L. [1]
In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group equipped with a word metric satisfying certain properties abstracted from classical hyperbolic geometry. The notion of a hyperbolic group was introduced and developed by ...
In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of n-dimensional hyperbolic geometry in which points are represented by points on the forward sheet S + of a two-sheeted hyperboloid in (n+1)-dimensional Minkowski space or by the displacement vectors from the origin to those points, and m ...
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
Many other mathematical objects have their origin in the hyperbola, such as hyperbolic paraboloids (saddle surfaces), hyperboloids ("wastebaskets"), hyperbolic geometry (Lobachevsky's celebrated non-Euclidean geometry), hyperbolic functions (sinh, cosh, tanh, etc.), and gyrovector spaces (a geometry proposed for use in both relativity and ...
Hyperbolic coordinates plotted on the Euclidean plane: all points on the same blue ray share the same coordinate value u, and all points on the same red hyperbola share the same coordinate value v. In mathematics, hyperbolic coordinates are a method of locating points in quadrant I of the Cartesian plane