Definition, Meaning & Synonyms | English word HYPERBOLIC

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Definitions of HYPERBOLIC

1. Of or relating to hyperbole.
2. Using hyperbole: exaggerated.
3. Of or pertaining to a hyperbola.
4. (vision, of a perceived color) Having a saturation exceeding 100%.
5. Indicates that the specified function is a hyperbolic function rather than a trigonometric function.
6. (mathematics, of a, metric space or a geometry) Having negative curvature or sectional curvature.
7. (geometry, topology, of an automorphism) Whose domain has two (possibly ideal) fixed points joined by a line mapped to itself by translation.
8. (topology) Of, pertaining to or in a hyperbolic space (a space having negative curvature or sectional curvature).

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Examples of Using HYPERBOLIC in a Sentence

• The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function.
• The Decca Navigator System was a hyperbolic radio navigation system that allowed ships and aircraft to determine their position by using radio signals from a dedicated system of static radio transmitters.
• Typical examples of entire functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic counterparts sinh and cosh, as well as derivatives and integrals of entire functions such as the error function.
• Loran-C is a hyperbolic radio navigation system that allows a receiver to determine its position by listening to low frequency radio signals that are transmitted by fixed land-based radio beacons.
• Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane.
• His work features mathematical objects and operations including impossible objects, explorations of infinity, reflection, symmetry, perspective, truncated and stellated polyhedra, hyperbolic geometry, and tessellations.
• Angle of parallelism, in hyperbolic geometry, the angle at one vertex of a right hyperbolic triangle that has two hyperparallel sides.
• The wave equation is a hyperbolic partial differential equation describing waves, including traveling and standing waves; the latter can be considered as linear superpositions of waves traveling in opposite directions.
• Bi-directional delay line, a numerical analysis technique used in computer simulation for solving ordinary differential equations by converting them to hyperbolic equations.
• In low-dimensional topology, Catalan's constant is 1/4 of the volume of an ideal hyperbolic octahedron, and therefore 1/4 of the hyperbolic volume of the complement of the Whitehead link.
• The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions.
• In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle.
• In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries.
• Sometime in the mid-to-late 1970s, William Thurston showed that the figure-eight was hyperbolic, by decomposing its complement into two ideal hyperbolic tetrahedra.
• Free groups first arose in the study of hyperbolic geometry, as examples of Fuchsian groups (discrete groups acting by isometries on the hyperbolic plane).
• It is a singular space (the equator is a singularity), but away from the singularities, it has constant negative Gaussian curvature and therefore is locally isometric to a hyperbolic plane.
• However, the two faithful spies Caleb and Joshua do not verify this report, leading some scholars to believe that the fearful reports from the other ten are hyperbolic and should not be taken literally.
• The paraboloid is hyperbolic if every other plane section is either a hyperbola, or two crossing lines (in the case of a section by a tangent plane).
• can be obtained by rearranging Stirling's extended formula and observing a coincidence between the resultant power series and the Taylor series expansion of the hyperbolic sine function.
• The theorem also has a well-known generalization to spherical and hyperbolic geometry, replacing the lengths in the ratios with their sines and hyperbolic sines, respectively.

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