hyperbolische

Hyperbolische is an adjective used in Dutch and German contexts to describe concepts related to a hyperbola, hyperbolic geometry, or the hyperbolic functions. The term derives from the Greek hyperbolē, meaning excess, and is connected to the geometric shape of a hyperbola as well as to mathematical structures that exhibit similar properties.

In geometry, hyperbolic geometry is a non-Euclidean geometry with constant negative curvature. It differs from Euclidean geometry in several fundamental ways: the parallel postulate is modified, and through a point not on a line there exist many lines that do not meet the original line. Triangles in hyperbolic geometry have angle sums less than 180 degrees. Several models are used to study it, such as the Poincaré disk model and the upper half-plane model, which provide concrete representations of hyperbolic distances and angles.

Hyperbolic functions are another core use of the term. The functions sinh, cosh, and tanh, along with their reciprocals, are defined via exponentials and have properties analogous to trigonometric functions, but tied to the unit hyperbola x^2 − y^2 = 1. They are widely used to solve certain differential equations, describe hyperbolic motion in physics, and appear in complex analysis and models of growth and decay.

Other related concepts include the hyperbolic paraboloid, a saddle-shaped surface, and various metric and geometric constructions that employ hyperbolic distance. In its broad sense, hyperbolische denotes ideas and objects associated with hyperbolas or spaces of negative curvature.