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One has a hyperboloid of revolution if and only if =. Otherwise, the axes are uniquely defined (up to the exchange of the x-axis and the y-axis). There are two kinds of hyperboloids. In the first case (+1 in the right-hand side of the equation): a one-sheet hyperboloid, also called a hyperbolic hyperboloid.
Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. 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 Cartesian coordinates .
The hyperbolic distance between two points on the hyperboloid can then be identified with the relative rapidity between the two corresponding observers. The model generalizes directly to an additional dimension: a hyperbolic 3-space three-dimensional hyperbolic geometry relates to Minkowski 4-space.
The hyperbolic paraboloid and the hyperboloid of one sheet are doubly ruled surfaces. The plane is the only surface which contains at least three distinct lines through each of its points ( Fuchs & Tabachnikov 2007 ).
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 ...
A hyperbolic link is a link in the 3-sphere whose complement (the space formed by removing the link from the 3-sphere) can be given a complete Riemannian metric of constant negative curvature, giving it the structure of a hyperbolic 3-manifold, a quotient of hyperbolic space by a group acting freely and discontinuously on it. The components of ...
The curve represents xy = 1. A hyperbolic angle has magnitude equal to the area of the corresponding hyperbolic sector, which is in standard position if a = 1. In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of xy = 1 in Quadrant I of the Cartesian plane.
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