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  2. Hyperbolic space - Wikipedia

    en.wikipedia.org/wiki/Hyperbolic_space

    Hyperbolic space, developed independently by Nikolai Lobachevsky, János Bolyai and Carl Friedrich Gauss, is a geometric space analogous to Euclidean space, but such that Euclid's parallel postulate is no longer assumed to hold. Instead, the parallel postulate is replaced by the following alternative (in two dimensions):

  3. Hyperbolic metric space - Wikipedia

    en.wikipedia.org/wiki/Hyperbolic_metric_space

    A definition of a -hyperbolic space is then a geodesic metric space all of whose geodesic triangles are -slim. This definition is generally credited to Eliyahu Rips . Another definition can be given using the notion of a C {\displaystyle C} -approximate center of a geodesic triangle: this is a point which is at distance at most C {\displaystyle ...

  4. Hyperboloid model - Wikipedia

    en.wikipedia.org/wiki/Hyperboloid_model

    Then n-dimensional hyperbolic space is a Riemannian space and distance or length can be defined as the square root of the scalar square. If the signature (+, −, −) is chosen, scalar square between distinct points on the hyperboloid will be negative, so various definitions of basic terms must be adjusted, which can be inconvenient.

  5. Hyperboloid - Wikipedia

    en.wikipedia.org/wiki/Hyperboloid

    In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes.A hyperboloid is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation.

  6. Hyperbolic motion - Wikipedia

    en.wikipedia.org/wiki/Hyperbolic_motion

    Hyperbolic motions are often taken from inversive geometry: these are mappings composed of reflections in a line or a circle (or in a hyperplane or a hypersphere for hyperbolic spaces of more than two dimensions). To distinguish the hyperbolic motions, a particular line or circle is taken as the absolute.

  7. Space form - Wikipedia

    en.wikipedia.org/wiki/Space_form

    In mathematics, a space form is a complete Riemannian manifold M of constant sectional curvature K. The three most fundamental examples are Euclidean n -space , the n -dimensional sphere , and hyperbolic space , although a space form need not be simply connected .

  8. List of hyperboloid structures - Wikipedia

    en.wikipedia.org/wiki/List_of_hyperboloid_structures

    A cluster of nine hyperbolic paraboloid concrete roofs Le Corbusier: Commissioned by the Netherlands electronics company Philips, the pavilion at the World's Fair Expo '58 in Brussels was designed by Le Corbusier to house a multimedia spectacle that celebrated postwar technological progress. Mürwik Wasserturm (Mürwik Water Tower) 1961 ...

  9. Hyperbolic manifold - Wikipedia

    en.wikipedia.org/wiki/Hyperbolic_manifold

    For > the hyperbolic structure on a finite volume hyperbolic -manifold is unique by Mostow rigidity and so geometric invariants are in fact topological invariants. One of these geometric invariants used as a topological invariant is the hyperbolic volume of a knot or link complement, which can allow us to distinguish two knots from each other ...