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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.
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 ...
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows .
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to −1. [1] It is homogeneous, and satisfies the stronger property of being a symmetric space.
Definition of a ruled surface: every point lies on a line. In geometry, a surface S in 3-dimensional Euclidean space is ruled (also called a scroll) if through every point of S, there is a straight line that lies on S.
In principle, a definition of prolate spheroidal coordinates could be degenerate. In other words, a single set of coordinates might correspond to two points in Cartesian coordinates; this is illustrated here with two black spheres, one on each sheet of the hyperboloid and located at (x, y, ±z). However, neither of the definitions presented ...
In differential geometry, this is a Lie transformation. In the corresponding solutions to the sine-Gordon equation, this deformation corresponds to a Lorentz Boost of the static 1-soliton solution. In some sources that use the hyperboloid model of the hyperbolic plane, the hyperboloid is referred to as a pseudosphere. [8]
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